Device, method, and program recording medium for control of air-fuel ratio of internal combustion engine

ABSTRACT

An apparatus for controlling the air-fuel ratio of an internal combustion engine to stably determine a highly reliable identified value of a parameter of a model of an exhaust system including a catalytic converter and to increase the purifying capability of the catalytic converter. An exhaust-side control unit  7   a  sequentially variably sets a dead time of an exhaust system E depending on the flow rate of an exhaust gas supplied to a catalytic converter  3 , and sequentially identifies the value of a parameter of a model of the exhaust system E which has a dead time element of the set dead time. The exhaust-side control unit  7   a  sequentially calculates a target air-fuel ratio KCMD to converge the output of an O 2  sensor  6  to a target value using the identified value of the parameter, and an engine-side control unit  7   b  manipulates the air-fuel ratio of the internal combustion engine  1  depending on the target air-fuel ratio KCMD. According to the algorithm of the process of identifying the parameter of the model of the exhaust system E, the value of a weighted parameter is variable set depending on the flow rate of the exhaust gas.

TECHNICAL FIELD

[0001] The present invention relates to an apparatus for and a method ofcontrolling the air-fuel ratio of an internal combustion engine, and arecording medium storing a program for controlling the air-fuel ratio ofan internal combustion engine.

BACKGROUND ART

[0002] There have already been proposed by the applicant of the presentapplication techniques for controlling the air-fuel ratio of an air-fuelmixture to be combusted by an internal combustion engine for convergingthe output of an exhaust gas sensor, e.g., an O₂ sensor (oxygenconcentration sensor), disposed downstream of a catalytic converter, toa predetermined target value (constant value) in order to achieve theappropriate purifying capability of the catalytic converter, such as athree-way catalyst or the like, disposed in the exhaust gas passage ofthe internal combustion engine (e.g., see Japanese laid-open patentpublication No. 11-324767, and Japanese laid-open patent publication No.2000-179385).

[0003] According to these techniques, an exhaust system ranging from aposition upstream of the catalytic converter to the O₂ sensor disposeddownstream of the catalytic converter is an object to be controlledwhich has an input quantity represented by the air-fuel ratio of theexhaust gas that enters the catalytic converter and an output quantityrepresented by the output of the O₂ sensor. A manipulated variable whichdetermines the input quantity of the exhaust system, e.g., a targetvalue for the input quantity of the exhaust system, is sequentiallygenerated by a feedback control process, or specifically an adaptivesliding mode control process, for converging the output of the O₂ sensorto the target value, and the air-fuel ratio of the air-fuel mixture tobe combusted by the internal combustion engine is controlled dependingon the manipulated variable.

[0004] Generally, the behavior and characteristics of the exhaust systemvary depending various factors including the operating state of theinternal combustion engine. The exhaust system including the catalyticconverter has a relatively long dead time.

[0005] According to the above techniques, the behavior of the exhaustsystem is modeled by regarding the exhaust system as a system forgenerating the output of the O₂ sensor from the air-fuel ratio of theexhaust gas that enters the catalytic converter via a dead time elementand a response delay element, and a parameter of the model of theexhaust system (a coefficient parameter relative to the dead timeelement and the response delay element) is sequentially identified usingsampled data of the output of the O₂ sensor and sampled data of theoutput of an air-fuel ratio sensor that is disposed upstream of thecatalytic converter for detecting the air-fuel ratio of the exhaust gasthat enters the catalytic converter. The manipulated variable issequentially generated using the identified value of the parameter ofthe model according to a feedback control process that is constructedbased on the model.

[0006] According to the above techniques, the process of identifying theparameter of the model of the exhaust system and the feedback controlprocess using the identified value of the parameter are carried out tocompensate for the effect of behavioral changes of the exhaust systemand smoothly perform the control process for converging the output ofthe O₂ sensor to the target value, or stated otherwise, an air-fuelratio control process for achieving an appropriate purifying capabilityof the catalytic converter.

[0007] According to the above techniques, basically, the dead time ofthe exhaust system is regarded as of a constant value, and a presetfixed dead time is used as the value of the dead time of the dead timeelement in the model of the exhaust system.

[0008] The inventors of the present application have found that theactual dead time of the exhaust system varies depending on the state,such as the rotational speed, of the internal combustion engine, and therange in which the dead time of the exhaust system is variable maybecome relatively large depending on the operating state of the internalcombustion engine. Consequently, depending on the operating state of theinternal combustion engine, an error between the model of the exhaustsystem and the behavior of the actual exhaust system may become large.Because of this error, an error and a variation of identified value ofthe parameter of the model of the exhaust system become large.

[0009] According to the above techniques, since a highly stable controlprocess such as an adaptive sliding mode control process is used as thefeedback control process for generating the manipulated variable, itbasically is possible to avoid a situation where the stability of thecontrol process for converging the output of the O₂ sensor to the targetvalue would significantly be impaired.

[0010] In circumstances where the error and variation of the identifiedvalue of the parameter of the model of the exhaust system is relativelylarge, however, when the manipulated variable is generated using theidentified value and the air-fuel ratio of the air-fuel mixture ismanipulated depending on the manipulated variable, the output of the O₂sensor tends to vary with respect to the target value, and the quickresponse of the control process converging the output of the O₂ sensorto the target value is liable to be lowered. This has presented anobstacle to efforts to further increase the purifying capability of thecatalytic converter.

[0011] The present invention has been made in view of the abovebackground. It is an object of the present invention to provide anapparatus for and a method of controlling the air-fuel ratio of aninternal combustion engine to stably determine a highly reliableidentified value of a parameter of a model of an exhaust systemincluding a catalytic converter and hence to increase the purifyingcapability of the catalytic converter in a system for manipulating theair-fuel ratio to converge the output of an exhaust gas sensor such asan O₂ sensor or the like disposed downstream of the catalytic converterto a predetermined target value to achieve an appropriate purifyingcapability of the catalytic converter. It is also an object of thepresent invention to provide a recording medium storing a program forcontrolling an air-fuel ratio appropriately with a computer.

DISCLOSURE OF THE INVENTION

[0012] According to the findings of the inventors of the presentapplication, the actual dead time of an exhaust system including acatalytic converter is closely related particularly to the flow rate ofan exhaust gas supplied to the catalytic converter such that the actualdead time of the exhaust system is longer as the flow rate of theexhaust gas is smaller (see the solid-line curve c in FIG. 4).Furthermore, the actual response delay time of the exhaust system islonger as the flow rate of the exhaust gas is smaller. The presentinvention has been made in view of such a phenomenon, and is availablein two aspects.

[0013] According to a first aspect of the present invention, there isprovided an apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, identifying means for sequentially identifying the value of apredetermined parameter of a predetermined model of an exhaust system,which ranges from a position upstream of the catalytic converter to theexhaust gas sensor and including the catalytic converter, for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of the exhaust gas sensor via at least a dead timeelement from the air-fuel ratio of the exhaust gas which enters thecatalytic converter, manipulated variable generating means forsequentially generating a manipulated variable to determine an air-fuelratio of the exhaust gas which enters the catalytic converter using theidentified value of the parameter of the model to converge the output ofthe exhaust gas sensor to a predetermined target value, and air-fuelratio manipulating means for manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable. The apparatus for controlling theair-fuel ratio according to the first aspect is characterized by flowrate data generating means for sequentially generating datarepresentative of a flow rate of the exhaust gas flowing through thecatalytic converter, and dead time setting means for variably setting aset dead time as the dead time of a dead time element of the model ofthe exhaust system depending on the value of the data generated by theflow rate data generating means, wherein the identifying meansidentifies the value of the parameter using the value of the set deadtime set by the dead time setting means.

[0014] According to the first aspect of the present invention, there isprovided a method of controlling the air-fuel ratio of an internalcombustion engine, comprising the steps of sequentially identifying thevalue of a predetermined parameter of a predetermined model of anexhaust system, which ranges from a position upstream of a catalyticconverter disposed in an exhaust passage of the internal combustionengine to an exhaust gas sensor disposed down-stream of the catalyticconverter for detecting the concentration of a particular component inan exhaust gas, and includes the catalytic converter, for expressing abehavior of the exhaust system which is regarded as a system forgenerating the output of the exhaust gas sensor via at least a dead timeelement from the air-fuel ratio of the exhaust gas which enters thecatalytic converter, and sequentially generating a manipulated variableto determine an air-fuel ratio of the exhaust gas which enters thecatalytic converter using the identified value of the parameter of themodel in order to converge the output of the exhaust gas sensor to apredetermined target value, the internal combustion engine havingair-fuel ratio manipulating means for manipulating the air-fuel ratio ofan air-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable. The method of controlling theair-fuel ratio according to the first aspect is characterized by thesteps of sequentially generating data representative of a flow rate ofthe exhaust gas flowing through the catalytic converter, andsequentially setting a set dead time as the dead time of a dead timeelement of the model of the exhaust system depending on the value of thedata representative of the flow rate of the exhaust gas, wherein thestep of identifying the parameter of the model of the exhaust systemidentifies the value of the parameter using the value of the set deadtime.

[0015] According to the first aspect of the present invention, there isprovided a recording medium readable by a computer and storing anair-fuel ratio control program for enabling the computer to perform aprocess of sequentially identifying the value of a predeterminedparameter of a predetermined model of an exhaust system, which rangesfrom a position upstream of a catalytic converter disposed in an exhaustpassage of the internal combustion engine to an exhaust gas sensordisposed downstream of the catalytic converter for detecting theconcentration of a particular component in an exhaust gas, and includesthe catalytic converter, for expressing a behavior of the exhaust systemwhich is regarded as a system for generating the output of the exhaustgas sensor via at least a dead time element from the air-fuel ratio ofthe exhaust gas which enters the catalytic converter, a process ofsequentially generating a manipulated variable to determine an air-fuelratio of the exhaust gas which enters the catalytic converter using theidentified value of the parameter of the model in order to converge theoutput of the exhaust gas sensor to a predetermined target value, and aprocess of manipulating the air-fuel ratio of an air-fuel mixture to becombusted by the internal combustion engine depending on the manipulatedvariable. The recording medium according to the first aspect ischaracterized in that the air-fuel ratio control program includes aprogram for enabling the computer to perform a process of sequentiallygenerating data representative of a flow rate of the exhaust gas flowingthrough the catalytic converter, and sequentially setting a value of theset dead time as the dead time of a dead time element of the model ofthe exhaust system depending on the value of the data representative ofthe flow rate of the exhaust gas, wherein the program for enabling thecomputer to identify the parameter of the model of the exhaust systemidentifies the parameter using the value of the set dead time.

[0016] According to the first aspect, the value of the set dead time ofthe exhaust system is established depending on the value of the datarepresentative of the flow rate of the exhaust gas. Therefore, the setdead time can be brought into conformity with the actual dead time ofthe exhaust system with accuracy. Basically, the set dead time isestablished such that it is greater as the flow rate of the exhaust gasrepresented by the above data is smaller.

[0017] According to the first aspect of the present invention, the valueof the set dead time, i.e., the value of the set dead time whichaccurately matches the actual dead time of the exhaust system, is usedas the dead time of the dead time element of the model for identifyingthe parameter of the model of the exhaust system. Therefore, matchingbetween the behavior of the model of the exhaust system and the behaviorof the actual exhaust system is increased, thus increasing thereliability of the identified value of the parameter of the model. Whenthe manipulated variable is generated using the identified value of theparameter, and the air-fuel ratio is manipulated depending on themanipulated variable, the accuracy and quick response of the controlprocess for converging the output of the exhaust gas sensor to thetarget value is increased. As a result, the purifying capability of thecatalytic converter is increased.

[0018] According to a second aspect of the present invention, there isprovided an apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, identifying means for sequentially identifying the value of apredetermined parameter of a predetermined model of an exhaust system,which ranges from a position upstream of the catalytic converter to theexhaust gas sensor and including the catalytic converter, for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of the exhaust gas sensor via at least a dead timeelement from the air-fuel ratio of the exhaust gas which enters thecatalytic converter, manipulated variable generating means forsequentially generating a manipulated variable to determine an air-fuelratio of the exhaust gas which enters the catalytic converter using theidentified value of the parameter of the model to converge the output ofthe exhaust gas sensor to a predetermined target value, and air-fuelratio manipulating means for manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable. The apparatus for controlling theair-fuel ratio according to the second aspect is characterized in thatthe identifying means comprises means for identifying the value of theparameter according to an algorithm for minimizing an error between theoutput of the exhaust gas sensor in the model of the exhaust system andan actual output of the exhaust gas sensor, and the apparatus is furthercharacterized by flow rate data generating means for sequentiallygenerating data representative of a flow rate of the exhaust gas flowingthrough the catalytic converter, and means for variably setting thevalue of a weighted parameter of the algorithm of the identifying meansdepending on the value of the data generated by the flow rate datagenerating means

[0019] Similarly, according to the second aspect of the presentinvention, there is provided a method of controlling the air-fuel ratioof an internal combustion engine, comprising the steps of sequentiallyidentifying the value of a predetermined parameter of a predeterminedmodel of an exhaust system, which ranges from a position upstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine to an exhaust gas sensor disposed downstream of thecatalytic converter for detecting the concentration of a particularcomponent in an exhaust gas, and includes the catalytic converter, forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of the exhaust gas sensor from theair-fuel ratio of the exhaust gas which enters the catalytic converter,and sequentially generating a manipulated variable to determine anair-fuel ratio of the exhaust gas which enters the catalytic converterusing the identified value of the parameter of the model in order toconverge the output of the exhaust gas sensor to a predetermined targetvalue, wherein the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine is manipulated depending on themanipulated variable. The method for controlling the air-fuel ratioaccording to the second aspect is characterized in that the step ofidentifying the parameter of the model of the exhaust system comprisesthe step of identifying the value of the parameter according to analgorithm for minimizing an error between the output of the exhaust gassensor in the model of the exhaust system and an actual output of theexhaust gas sensor, and the method further comprises the steps ofsequentially generating data representative of a flow rate of theexhaust gas flowing through the catalytic converter, and variablysetting the value of a weighted parameter of the algorithm foridentifying the parameter of the model depending on the value of thedata representative of the flow rate of the exhaust gas.

[0020] According to the second aspect of the present invention, there isprovided a recording medium readable by a computer and storing anair-fuel ratio control program for enabling the computer to perform aprocess of sequentially identifying the value of a predeterminedparameter of a predetermined model of an exhaust system, which rangesfrom a position upstream of a catalytic converter disposed in an exhaustpassage of the internal combustion engine to an exhaust gas sensordisposed downstream of the catalytic converter for detecting theconcentration of a particular component in an exhaust gas, and includesthe catalytic converter, for expressing a behavior of the exhaust systemwhich is regarded as a system for generating the output of the exhaustgas sensor from the air-fuel ratio of the exhaust gas which enters thecatalytic converter, a process of sequentially generating a manipulatedvariable to determine an air-fuel ratio of the exhaust gas which entersthe catalytic converter using the identified value of the parameter ofthe model in order to converge the output of the exhaust gas sensor to apredetermined target value, and a process of manipulating the air-fuelratio of an air-fuel mixture to be combusted by the internal combustionengine depending on the manipulated variable. The recording mediumaccording to the second aspect is characterized in that the program ofthe air-fuel ratio control program for enabling the computer to performthe process of identifying the value of the parameter of the model ofthe exhaust system identifies the value of the parameter according to analgorithm for minimizing an error between the output of the exhaust gassensor in the model of the exhaust system and an actual output of theexhaust gas sensor, and the air-fuel ratio control program includes aprogram for enabling the computer to perform a process of sequentiallygenerating data representative of a flow rate of the exhaust gas flowingthrough the catalytic converter, and a process of variably setting thevalue of a weighted parameter of the algorithm for identifying theparameter of the model depending on the value of the data representativeof the flow rate of the exhaust gas.

[0021] The findings of the inventors of the present application indicatethat as the actual dead time and response delay time of the exhaustsystem are longer, the identified value of the parameter of the model ofthe exhaust system is liable to suffer variations and errors, tending toimpair the quick response of the control process for converging theoutput of the exhaust gas sensor to the target value. If an algorithmsuch as a method of weighted least squares is used as the algorithm foridentifying the value of the parameter of the model of the exhaustsystem, then it is possible to reduce variations and errors of theidentified value of the parameter of the model of the exhaust system byadjusting the value of a weighted parameter of the algorithm.

[0022] According to the second aspect of the present invention,therefore, an algorithm such as a method of weighted least squares isused to identify the value of the parameter of the model of the exhaustsystem, and the value of the weighted parameter of the algorithm isvariably set depending on the value of the data representative of theflow rate of the exhaust gas. The value of the weighted parameter canthus be adjusted so as to match the actual dead time and response delaycharacteristics of the exhaust system. As a result, it is possible toreduce variations and errors of the identified value of the parameter ofthe model of the exhaust system and stably obtain a highly reliableidentified value, and hence the quick response and accuracy of thecontrol process for converging the output of the exhaust gas sensor tothe target value is increased. As a result, the purifying capability ofthe catalytic converter can be increased.

[0023] According to the second aspect of the present invention, themodel of the exhaust system may include at least a dead time element(e.g., it may include both a dead time element and a response delayelement). However, the model of the exhaust system may include only aresponse delay element without a dead time element.

[0024] The apparatus for controlling the air-fuel ratio, the method ofcontrolling the air-fuel ratio, and the recording medium storing theair-fuel ratio control program according to the present invention mayhave the arrangements of both the first and second aspects for furtherincreasing the accuracy and quick response of the control process forconverging the output of the exhaust gas sensor to the target value andhence further increasing the purifying capability of the catalyticconverter.

[0025] According to the first and second aspects of the presentinvention, the manipulated variable may be a target value for theair-fuel ratio (target air-fuel ratio) of the exhaust gas that entersthe catalytic converter, a corrective amount for the amount of fuelsupplied to the internal combustion engine, or the like. If themanipulated variable is a target air-fuel ratio, then it is preferableto provide an air-fuel ratio sensor upstream of the catalytic converterfor detecting the air-fuel ratio of the exhaust gas that enters thecatalytic converter, and manipulate the air-fuel ratio of an air-fuelmixture to be combusted by the internal combustion engine according to afeedback control process for converging the output of the air-fuel ratiosensor (the detected value of the air-fuel ratio) to the target air-fuelratio.

[0026] According to the first and second aspects of the presentinvention, more specifically, it is possible to identify the value ofthe parameter of the model of the exhaust system according the algorithmof a sequential method of weighted least squares (the algorithm of amethod of weighted least squares in the second embodiment), using thedata representative of the air-fuel ratio of the exhaust gas that entersthe catalytic converter (hereinafter also referred to as“upstream-of-catalyst air-fuel ratio”), the data of the output of theexhaust gas sensor, and the value of the set dead time of the model ofthe exhaust system. In any of the aspects, the data of the manipulatedvariable can be used as the data representative of theupstream-of-catalyst air-fuel ratio since the upstream-of-catalystair-fuel ratio is determined by the manipulated variable. However, it ispreferable to provide an air-fuel ratio sensor for detecting theupstream-of-catalyst air-fuel ratio up-stream of the catalytic converterand use the data of an output of the air-fuel ratio sensor as datarepresentative of the upstream-of-catalyst air-fuel ratio.

[0027] The model of the exhaust system according to the first and secondaspects of the present invention should preferably be a model whichexpresses the data of the output of the exhaust gas sensor in each givencontrol cycle with the data of the output of the exhaust gas sensor in apast control cycle prior to the control cycle and the datarepresentative of the upstream-of-catalyst air-fuel ratio (the data ofthe output of the air-fuel ratio sensor, the data of the manipulatedvariable, or the like) in a control cycle prior to the set dead time ofthe exhaust system, for example. Stated otherwise, the model shouldpreferably be an autoregressive model where the upstream-of-catalystair-fuel ratio as an input quantity to the exhaust system has a deadtime (the set dead time of the exhaust system), for example. Theparameter of the model comprises a coefficient relative to the data(autoregressive term relative to an output quantity of the exhaustsystem) of the output of the exhaust gas sensor in the past controlcycle, or a coefficient relative to the data (input quantity to theexhaust system) representative of the upstream-of-catalyst air-fuelratio.

[0028] In the apparatus for controlling the air-fuel ratio according tothe first and second aspects of the present invention, the identifyingmeans should preferably determine the identified value of the parameterof the model of the exhaust system by limiting the identified value to avalue within a predetermined range depending on the value of the datagenerated by the flow rate data generating means.

[0029] Likewise, in the method of controlling the air-fuel ratioaccording to the first and second aspects of the present invention, thestep of identifying the parameter of the model of the exhaust systemshould preferably determine the identified value of the parameter of themodel of the exhaust system by limiting the identified value to a valuewithin a predetermined range depending on the value of the datarepresentative of the flow rate of the exhaust gas.

[0030] Furthermore, in the recording medium according to the first andsecond aspects, the program of the air-fuel ratio control program forenabling the computer to perform the process of identifying the value ofthe parameter of the model of the exhaust system should preferablydetermine the identified value of the parameter of the model of theexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data representative ofthe flow rate of the exhaust gas.

[0031] Specifically, the identified value of the parameter which issuitable for generating the manipulated variable capable of convergingthe output of the exhaust gas sensor smoothly to the target value isgenerally affected by the actual dead time of the exhaust system becauseof the effect of the flow rate of the exhaust gas. According to thepresent invention, the identified value is limited to a value within apredetermined range that is determined depending on the value of thedata representative of the flow rate of the exhaust gas. It is thuspossible to determine the identified value suitable for generating themanipulated variable capable of converging the output of the exhaust gassensor smoothly to the target value.

[0032] If there are a plurality of parameters to be identified of themodel of the exhaust system, then the predetermined range within whichto limit the identified values of those parameters may be a range foreach of the identified values of those parameters or a range for acombination of the identified values of those parameters. For example,if the model of the exhaust system is an autoregressive model and itsautoregressive terms include primary and secondary autoregressive terms(which correspond to the response delay element of the exhaust system),then it is preferable to limit a combination of the identified values oftwo parameters relative to the respective autoregressive terms within apredetermined range (specifically, a predetermined area on a coordinateplane having the values of the two parameters as representing twocoordinate axes). The identified value of a parameter relative to theup-stream-of-catalyst air-fuel ratio of the autoregressive model shouldpreferably be limited to a value within a pre-determined range (a rangehaving upper and lower limit values).

[0033] In the first and second aspects of the present invention, thefeedback control process for generating the manipulated variable shouldpreferably be an adaptive control process, or more specifically, asliding mode control process. The sliding mode control process may be anordinary sliding mode control process based on a control law relative toan equivalent control input and a reaching law, but should preferably bean adaptive sliding mode control process with an adaptive law (adaptivealgorithm) added to those control laws.

BRIEF DESCRIPTION OF THE DRAWINGS

[0034]FIG. 1 is a block diagram of an overall system arrangement of anapparatus for controlling the air-fuel ratio of an internal combustionengine according to a first embodiment of the present invention;

[0035]FIG. 2 is a diagram showing output characteristics of an O₂ sensorused in the apparatus shown in FIG. 1;

[0036]FIG. 3 is a block diagram showing a basic arrangement of a targetair-fuel ratio generation processor of the apparatus shown in FIG. 1;

[0037]FIG. 4 is a diagram illustrative of a process performed by a deadtime setting means of the target air-fuel ratio generation processorshown in FIG. 3;

[0038]FIG. 5 is a diagram illustrative of a process performed by anidentifier of the target air-fuel ratio generation processor shown inFIG. 3;

[0039]FIG. 6 is a diagram with respect to a sliding mode controller ofthe target air-fuel ratio generation processor shown in FIG. 3;

[0040]FIG. 7 is a block diagram showing a basic arrangement of anadaptive controller of the apparatus shown in FIG. 1;

[0041]FIG. 8 is a flowchart of a processing sequence of an engine-sidecontrol unit (7 b) of the apparatus shown in FIG. 1;

[0042]FIG. 9 is a flowchart of a subroutine of the flowchart shown inFIG. 8;

[0043]FIG. 10 is a flowchart of an overall processing sequence of anexhaust-side control unit (7 a) of the apparatus shown in FIG. 1;

[0044]FIGS. 11 and 12 are flowcharts of sub-routines of the flowchartshown in FIG. 10;

[0045]FIGS. 13 through 15 are diagrams illustrating partial processes ofthe flowchart shown in FIG. 12;

[0046]FIG. 16 is a flowchart of a subroutine of the flowchart shown inFIG. 12; and

[0047]FIG. 17 is a flowchart of a subroutine of the flowchart shown inFIG. 10.

BEST MODE FOR CARRYING OUT THE INVENTION

[0048] A first embodiment of the present invention will be describedbelow with reference to FIGS. 1 through 17. The present embodiment is anembodiment relating to the first aspect of the present invention andalso an embodiment relating to the second aspect.

[0049]FIG. 1 shows in block form an overall system arrangement of anapparatus for controlling the air-fuel ratio of an internal combustionengine according to the present embodiment. As shown in FIG. 1, aninternal combustion engine 1 such as a four-cylinder internal combustionengine is mounted as a propulsion source on an automobile or a hybridvehicle, for example. When a mixture of fuel and air is combusted ineach cylinder of the internal combustion engine 1, an exhaust gas isgenerated and emitted from each cylinder into a common discharge pipe 2(exhaust passage) positioned near the internal combustion engine 1, fromwhich the exhaust gas is discharged into the atmosphere. Two three-waycatalytic converters 3, 4, each comprising a three-way catalyst forpurifying the exhaust gas, are mounted in the common exhaust pipe 2 atsuccessively downstream locations thereon. The downstream catalyticconverter 4 may be dispensed with.

[0050] The system according to the present embodiment serves to controlthe air-fuel ratio of the internal combustion engine 1 (or moreaccurately, the air-fuel ratio of the mixture of fuel and air to becombusted by the internal combustion engine 1, the same applieshereinafter) in order to achieve an optimum purifying capability of thecatalytic converter 3. In order to perform the above control process,the system according to the present embodiment has an air-fuel ratiosensor 5 mounted on the exhaust pipe 2 upstream of the catalyticconverter 3 (or more specifically at a position where exhaust gases fromthe cylinders of the internal combustion engine 1 are put together), anO₂ sensor nal combustion engine 1 are put together), an O₂ sensor(oxygen concentration sensor) 6 mounted as an exhaust gas sensor on theexhaust pipe 2 downstream of the catalytic converter 3 (upstream of thecatalytic converter 4), and a control unit 7 for carrying out a controlprocess (described later on) based on outputs (detected values) from thesensors 5, 6. The control unit 7 is supplied with outputs from varioussensors (not shown) for detecting operating conditions of the internalcombustion engine 1, including a engine speed sensor, an intake pressuresensor, a coolant temperature sensor, etc.

[0051] The O₂ sensor 6 comprises an ordinary O₂ sensor for generating anoutput VO2/OUT having a level depending on the oxygen concentration inthe exhaust gas that has passed through the catalytic converter 3 (anoutput representing a detected value of the oxygen concentration of theexhaust gas). The oxygen concentration in the exhaust gas iscommensurate with the air-fuel ratio of an air-fuel mixture which, whencombusted, produces the exhaust gas. The output VO2/OUT from the O₂sensor 6 will change with high sensitivity substantially linearly inproportion to the oxygen concentration in the exhaust gas, with theair-fuel ratio corresponding to the oxygen concentration in the exhaustgas being in a relatively narrow range Δ close to a stoichiometricair-fuel ratio, as indicated by the solid-line curve a in FIG. 2. Atoxygen concentrations corresponding to air-fuel ratios outside of therange Δ, the output VO2/OUT from the O₂ sensor 6 is saturated and is ofa substantially constant level.

[0052] The air-fuel ratio sensor 5 generates an output KACT representinga detected value of the air-fuel ratio of the exhaust gas that entersthe catalytic converter 3 (more specifically, an air-fuel ratio which isrecognized from the concentration of oxygen in the exhaust gas thatenters the catalytic converter 3). The air-fuel ratio sensor 5 comprisesa wide-range air-fuel ration sensor disclosed in Japanese laid-openpatent publication No. 4-369471 by the applicant of the presentapplication. As indicated by the solid-line curve b in FIG. 2, theair-fuel ratio sensor 5 generates an output KACT whose level isproportional to the concentration of oxygen in the exhaust gas in awider range than the O₂ sensor 5. In the description which follows, theair-fuel ratio sensor 5 will be referred to as “LAF sensor 5”, and theair-fuel ratio of the exhaust gas that enters the catalytic converter 3as “upstream-of-catalyst air-fuel ratio”.

[0053] The control unit 7 comprises a microcomputer, and has anexhaust-side control unit 7 a for performing, in pre-determined controlcycles, a process of sequentially generating a target air-fuel ratioKCMD for the upstream-of-catalyst air-fuel ratio (which is also a targetvalue for the output KACT of the LAF sensor 5) as a manipulated variablefor determining the upstream-of-catalyst air-fuel ratio, and anengine-side control unit 7 b for sequentially carryout out, inpredetermined control cycles, a process of manipulating theupstream-of-catalyst air-fuel ratio by adjusting an amount of fuelsupplied to the internal combustion engine 1 depending on the targetair-fuel ratio KCMD. These control units 7 a, 7 b correspondrespectively to a manipulated variable generating means and an air-fuelratio manipulating means according to the present invention. The controlunit 7 has a program stored in advance in a ROM for enabling a CPU toperform the control processes of the exhaust-side control unit 7 a andthe engine-side control unit 7 b as described later on. The control unit7 has the ROM as a recording medium according to the present invention.

[0054] In the present embodiments, the control cycles in which thecontrol units 7 a, 7 b perform their respective processing sequences aredifferent from each other. Specifically, the control cycles of theprocessing sequence of the exhaust-side control unit 7 a have apredetermined fixed period (e.g., ranging from 30 to 100 ms) in view ofthe relatively long dead time present in an exhaust system E (describedlater on) including the catalytic converter 3, calculating loads, etc.The control cycles of the processing sequence of the engine-side controlunit 7 b have a period in synchronism with the crankshaft angle period(so-called TDC) of the internal combustion engine 1 because the processof adjusting the amount of fuel supplied to the internal combustionengine 1 needs to be in synchronism with combustion cycles of theinternal combustion engine 1. The period of the control cycles of theexhaust-side control unit 7 a is longer than the crankshaft angle period(TDC) of the internal combustion engine 1.

[0055] The processing sequences of the control units 7 a, 7 b will bedescribed below. The engine-side control unit 7 b has, as its functions,a basic fuel injection quantity calculator 8 for determining a basicfuel injection quantity Tim to be injected into the internal combustionengine 1, a first correction coefficient calculator 9 for determining afirst correction coefficient KTOTAL to correct the basic fuel injectionquantity Tim, and a second correction coefficient calculator 10 fordetermining a second correction coefficient KCMDM to correct the basicfuel injection quantity Tim.

[0056] The basic fuel injection quantity calculator 8 determines areference fuel injection quantity (an amount of supplied fuel) from therotational speed NE and intake pressure PB of the internal combustionengine 1 using a predetermined map, and corrects the determinedreference fuel injection quantity depending on the effective openingarea of a throttle valve (not shown) of the internal combustion engine1, thereby calculating a basic fuel injection quantity Tim.

[0057] The first correction coefficient KTOTAL determined by the firstcorrection coefficient calculator 9 serves to correct the basic fuelinjection quantity Tim in view of an exhaust gas recirculation ratio ofthe internal combustion engine 1 (the proportion of an exhaust gascontained in an air-fuel mixture introduced into the internal combustionengine 10), an amount of purged fuel supplied to the internal combustionengine 1 when a canister (not shown) is purged, a coolant temperature,an intake temperature, etc. of the internal combustion engine 1.

[0058] The second correction coefficient KCMDM determined by the secondcorrection coefficient calculator 10 serves to correct the basic fuelinjection quantity Tim in view of the charging efficiency of an intakeair due to the cooling effect of fuel flowing into the internalcombustion engine 1 depending on a target air-fuel ratio KCMD which isdetermined by the exhaust-side control unit 7 a, as described later on.

[0059] The engine-side control unit 7 b corrects the basic fuelinjection quantity Tim with the first correction coefficient KTOTAL andthe second correction coefficient KCMDM by multiplying the basic fuelinjection quantity Tim by the first correction coefficient KTOTAL andthe second correction coefficient KCMDM, thus producing a demand fuelinjection quantity Tcyl for the internal combustion engine 1.

[0060] Specific details of processes for calculating the basic fuelinjection quantity Tim, the first correction coefficient KTOTAL, and thesecond correction coefficient KCMDM are disclosed in detail in Japaneselaid-open patent publication No. 5-79374 by the applicant of the presentapplication, and will not be described below.

[0061] The engine-side control unit 7 b also has, in addition to theabove functions, a feedback controller 14 for feedback-controlling theair-fuel ratio of the internal combustion engine 1 by adjusting a fuelinjection quantity of the internal combustion engine 1 so as to convergethe output KACT of the LAF sensor 5 (the detected value of theup-stream-of-catalyst air-fuel ratio) to the target air-fuel ratio KCMDwhich is sequentially calculated by the exhaust-side control unit 7 a(to be described in detail later).

[0062] The feedback controller 14 comprises a general feedbackcontroller 15 for feedback-controlling a total air-fuel ratio of thecylinders of the internal combustion engine 1 and a local feedbackcontroller 16 for feedback-controlling an air-fuel ratio of each of thecylinders of the internal combustion engine 1.

[0063] The general feedback controller 15 sequentially determines afeedback correction coefficient KFB to correct the demand fuel injectionquantity Tcyl (by multiplying the demand fuel injection quantity Tcyl)so as to converge the output KACT from the LAF sensor 5 to the targetair-fuel ratio KCMD. The general feedback controller 15 comprises a PIDcontroller 17 for generating a feedback manipulated variable KLAF as thefeedback correction coefficient KFB depending on the difference betweenthe output KACT from the LAF sensor 5 and the target air-fuel ratio KCMDaccording to a known PID control process, and an adaptive controller 18(indicated by “STR” in FIG. 1) for adaptively determining a feedbackmanipulated variable KSTR for determining the feedback correctioncoefficient KFB in view of changes in operating state of the internalcombustion engine 1 or characteristic changes thereof from the outputKACT from the LAF sensor 5 and the target air-fuel ratio KCMD.

[0064] In the present embodiment, the feedback manipulated variable KLAFgenerated by the PID controller 17 is of “1” and can be used directly asthe feedback correction co-efficient KFB when the output KACT (thedetected value of the upstream-of-catalyst air-fuel ratio) from the LAFsensor 5 coincides with the target air-fuel ratio KCMD. The feedbackmanipulated variable KSTR generated by the adaptive controller 18becomes the target air-fuel ratio KCMD when the output KACT from the LAFsensor 5 is equal to the target air-fuel ratio KCMD. A feedbackmanipulated variable kstr (=KSTR/KCMD) which is produced by dividing thefeedback manipulated variable KSTR by the target air-fuel ratio KCMDwith a divider 19 can be used as the feedback correction co-efficientKFB.

[0065] The feedback manipulated variable KLAF generated by the PIDcontroller 17 and the feedback manipulated variable kstr which isproduced by dividing the feedback manipulated variable KSTR from theadaptive controller 18 by the target air-fuel ratio KCMD are selectedone at a time by a switcher 20 of the general feedback controller 15. Aselected one of the feedback manipulated variable KLAF and the feedbackmanipulated variable KSTR is used as the feedback correction coefficientKFB. The demand fuel injection quantity Tcyl is corrected by beingmultiplied by the feedback correction coefficient KFB. Details of thegeneral feedback controller 15 (particularly, the adaptive controller18) will be described later on.

[0066] The local feedback controller 16 comprises an observer 21 forestimating real air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respectivecylinders from the output KACT from the LAF sensor 5, and a plurality ofPID controllers 22 (as many as the number of the cylinders) fordetermining respective feedback correction coefficients #nKLAF for fuelinjection quantities for the cylinders from the respective real air-fuelratios #nA/F estimated by the observer 21 according to a PID controlprocess so as to eliminate variations of the air-fuel ratios of thecylinders.

[0067] Briefly stated, the observer 21 estimates a real air-fuel ratio#nA/F of each of the cylinders as follows: A system from the internalcombustion engine 1 to the LAF sensor 5 (where the exhaust gases fromthe cylinders are combined) is considered to be a system for generatingan up-stream-of-catalyst air-fuel ratio detected by the LAF sensor 5from a real air-fuel ratio #nA/F of each of the cylinders, and ismodeled in view of a detection response delay (e.g., a time lag of firstorder) of the LAF sensor 5 and a chronological contribution of theair-fuel ratio of each of the cylinders to the upstream-of-catalystair-fuel ratio. Based on the modeled system, a real air-fuel ratio #nA/Fof each of the cylinders is estimated from the output KACT from the LAFsensor 5.

[0068] Details of the observer 21 are disclosed in Japanese laid-openpatent publication No. 7-83094 by the applicant of the presentapplication, and will not be described below.

[0069] Each of the PID controllers 22 of the local feedback controller16 divides the output signal KACT from the LAF sensor 5 by an averagevalue of the feedback correction coefficients #nKLAF determined by therespective PID controllers 22 in a preceding control cycle to produce aquotient value, and uses the quotient value as a target air-fuel ratiofor the corresponding cylinder. Each of the PID controllers 22 thendetermines a feedback correction coefficient #nKLAF in a present controlcycle so as to eliminate any difference between the target air-fuelratio and the corresponding real air-fuel ratio #nA/F determined by theobserver 21.

[0070] The local feedback controller 16 multiplies a value, which hasbeen produced by multiplying the demand fuel injection quantity Tcyl bythe selected feedback correction coefficient KFB produced by the generalfeedback controller 15, by the feedback correction coefficient #nKLAFfor each of the cylinders, thereby determining an output fuel injectionquantity #nTout (n=1, 2, 3, 4) for each of the cylinders.

[0071] The output fuel injection quantity #nTout thus determined foreach of the cylinders is corrected for accumulated fuel particles onintake pipe walls of the internal combustion engine 1 by a fuelaccumulation corrector 23 in the engine-side control unit 7 b. Thecorrected output fuel injection quantity #nTout is applied to each offuel injectors (not shown) of the internal combustion engine 1, whichinjects fuel into each of the cylinders with the corrected output fuelinjection quantity #nTout.

[0072] The correction of the output fuel injection quantity in view ofaccumulated fuel particles on intake pipe walls is disclosed in detailin Japanese laid-open patent publication No. 8-21273 by the applicant ofthe present application, and will not be described in detail below. Asensor output selector 24 shown in FIG. 1 serves to select the outputKACT from the LAF sensor 5, which is suitable for the estimation of areal air-fuel ratio #nA/F of each cylinder with the observer 21,depending on the operating state of the internal combustion engine 1.Details of the sensor output selector 24 are disclosed in detail inJapanese laid-open patent publication No. 7-259588 or U.S. Pat. No.5,540,209 by the applicant of the present application, and will not bedescribed in detail below.

[0073] The exhaust-side control unit 7 a has a subtractor 11 forsequentially determining a difference kact (=KACT−FLAF/BASE) between theoutput KACT from the LAF sensor 5 and a predetermined air-fuel ratioreference value FLAF/BASE and a subtractor 12 for sequentiallydetermining a difference VO2 (=VO2/OUT−VO2/TARGET) between the out-putVO2/OUT from the O₂ sensor 6 and a target value VO2/TARGET therefor.

[0074] The target value VO2/TARGET for the output VO2/OUT from the O₂sensor 6 is a predetermined value as an output value of the O₂ sensor 6in order to achieve an optimum purifying capability of the catalyticconverter 3 (specifically, purification ratios for NOx, HC, CO, etc. inthe exhaust gas), and is an output value that can be generated by the O₂sensor 6 in a situation where the air-fuel ratio of the exhaust gas ispresent in the range Δ close to a stoichiometric air-fuel ratio as shownin FIG. 2. In the present embodiment, the reference value FLAF/BASE withrespect to the output KACT from the LAF sensor 5 is set to a“stoichiometric air-fuel ratio” (constant value).

[0075] In the description which follows, the differences kact, VO2determined respectively by the subtractors 11, 12 are referred to as adifferential output kact of the LAF sensor 5 and a differential outputVO2 of the O₂ sensor 6, respectively.

[0076] The exhaust-side control unit 7 a also has a target air-fuelratio generation processor 13 for sequentially calculating the targetair-fuel ratio KCMD (the target value for the upstream-of-catalystair-fuel ratio) based on the data of the differential outputs kact, VO2used respectively as the data of the output from the LAF sensor 5 andthe output of the O₂ sensor 6.

[0077] The target air-fuel ratio generation processor 13 serves tocontrol, as an object control system, an exhaust system (denoted by E inFIG. 1) including the catalytic converter 3, which ranges from the LAFsensor 5 to the O₂ sensor 6 along the exhaust pipe 2. The targetair-fuel ratio generation processor 13 sequentially determines thetarget air-fuel ratio KCMD for the internal combustion engine 1 so as toconverge (settle) the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET therefor according to a sliding mode control process(specifically an adaptive sliding mode control process) in view of adead time present in the exhaust system E, a dead time present in anair-fuel ratio manipulating system comprising the internal combustionengine 1 and the engine-side control unit 7 b, and behavioral changes ofthe exhaust system E.

[0078] In order to carry out the control process of the target air-fuelratio generation processor 13, according to present embodiment, theexhaust system E is regarded as a system for generating the outputVO2/OUT of the O₂ sensor 6 from the output KACT of the LAF sensor 5 (theupstream-of-catalyst air-fuel ratio detected by the LAF sensor 5) via adead time element and a response delay element, and a model isconstructed for expressing the behavior of the exhaust system E. Theair-fuel ratio manipulating system comprising the internal combustionengine 1 and the engine-side control unit 7 b is regarded as a systemfor generating the output KACT of the LAF sensor 5 from the targetair-fuel ratio KCMD via a dead time element, and a model is constructedfor expressing the behavior of the air-fuel ratio manipulating system.

[0079] With respect to the model of the exhaust system E (hereinafterreferred to as “exhaust system model”), the behavior of the exhaustsystem E is expressed by an autoregressive model of a discrete timesystem according to the equation (1) shown below (specifically, anautoregressive model having a dead time in the differential output kactas the input quantity of the exhaust system E), using the differentialoutput kact (=KACT−FLAF/BASE) from the LAF sensor 5 as the inputquantity of the exhaust system E and the differential output VO2(=VO2/OUT−VO2/TARGET) from the O₂ sensor 6 as the output quantity of theexhaust system E, instead of the output KACT of the LAF sensor 5 and theoutput VO2/OUT of the O₂ sensor 6.

VO 2(k+1)=a 1·VO 2(k)+a 2·VO 2(k−1)+b 1·kact(k−d 1)  (1)

[0080] In the equation (1), “k” represents the ordinal number of adiscrete-time control cycle of the exhaust-side control unit 7 a, and“d1” the dead time of the exhaust system E (more specifically, the deadtime required until the upstream-of-catalyst air-fuel ratio detected ateach point of time by the LAF sensor 5 is reflected in the outputVO2/OUT of the O₂ sensor 6) as represented by the number of controlcycles. The actual dead time of the exhaust system E is closely relatedto the flow rate of the exhaust gas supplied to the catalytic converter3, and is basically longer as the flow rate of the exhaust gas issmaller. This is because as the flow rate of the exhaust gas is smaller,the time required for the exhaust gas to pass through the catalyticconverter 3 is longer. In the present embodiment, the flow rate of theexhaust gas supplied to the catalytic converter 3 is sequentiallygrasped, and the value of the dead time d1 in the exhaust system modelaccording to the equation (1) is variably set (the set value of the deadtime d1 will hereinafter be referred to as “set dead time d1”).

[0081] The first and second terms of the right side of the equation (1)correspond to a response delay element of the exhaust system E, thefirst term being a primary auto-regressive term and the second termbeing a secondary auto-regressive term. In the first and second terms,“a1”, “a2” represent respective gain coefficients of the primaryauto-regressive term and the secondary autoregressive term. Statedotherwise, these gain coefficients a1, a2 are relative to thedifferential output VO2 of the O₂ sensor 6 as an output quantity of theexhaust system E.

[0082] The third term of the right side of the equation (1) correspondsto a dead time element of the exhaust system E, and represents thedifferential output kact of the LAF sensor 5 as an input quantity of theobject exhaust system E, including the dead time d1 of the exhaustsystem E. In the third term, “b1” represents a gain coefficient relativeto the dead time element (an input quantity having the dead time d1).

[0083] These gain coefficients “a1”, “a2”, “b1” are parameters to be setto certain values for defining the behavior of the model of the exhaustsystem E, and are sequentially identified by an identifier which will bedescribed later on according to the present embodiment.

[0084] The exhaust system model expressed by the equation (1) thusexpresses the differential output VO2(k+1) of the O₂ sensor as the inputquantity of the exhaust system E in each control cycle of theexhaust-side control unit 7 a, with the differential outputs VO2(k),VO2(k−1) in past control cycles prior to that control cycle and thedifferential output kact(k-d1) of the LAF sensor 5 as the input quantity(upstream-of-catalyst air-fuel ratio) of the exhaust system E in acontrol cycle prior to the dead time d1 of the exhaust system E.

[0085] With respect to the model of the air-fuel ratio manipulatingsystem comprising the internal combustion engine 1 and the engine-sidecontrol unit 7 b (hereinafter referred to as “air-fuel ratiomanipulating system model”), the difference kcmd (=KCMD−FLAF/BASE,hereinafter referred to as “target differential air-fuel ratio kcmd”)between the target air-fuel ratio KCMD and the air-fuel ratio referencevalue FLAF/BASE is regarded as an input quantity of the air-fuel ratiomanipulating system, the differential output kact of the LAF sensor 5 asan output quantity of the air-fuel ratio manipulating system, and thebehavior of the air-fuel ratio manipulating system model is expressed bya model according to the following equation (2):

kact(k)=kcmd (k−d 2)  (2)

[0086] In the equation (2), “d2” represents the dead time of theair-fuel ratio manipulating system (more specifically, the dead timerequired until the target air-fuel ratio KCMD at each point of time isreflected in the output KACT of the LAF sensor 5) in terms of the numberof control cycles of the exhaust-side control unit 7 a. The actual deadtime of the air-fuel ratio manipulating system is closely related to theflow rate of the exhaust gas supplied to the catalytic converter 3, aswith the dead time of the exhaust system E, and is basically longer asthe flow rate of the exhaust gas is smaller. This because as the flowrate of the exhaust gas is smaller, the rotational speed of the internalcombustion engine 1 is lower (the crankshaft angle period is longer),and the period of the control cycles of the engine-side control unit 7 bof the air-fuel ratio manipulating system is longer. In the presentembodiment, therefore, the flow rate of the exhaust gas supplied to thecatalytic converter 3 is sequentially recognized, and the value of thedead time t2 in the air-fuel ratio manipulating system according to theequation (2) is variably set (the set value of the dead time d2 willhereinafter be referred to as “set dead time d2”).

[0087] The air-fuel ratio manipulating system model expressed by theequation (2) regards the air-fuel ratio manipulating system as a systemwherein the differential output kact of the LAF sensor 5 as the outputquantity (up-stream-of-catalyst air-fuel ratio) of the air-fuel ratiomanipulating system coincides with the target differential air-fuelratio kcmd as the input quantity of the air-fuel ratio manipulatingsystem at a time prior to the dead time t2 in the air-fuel ratiomanipulating system, and expresses the behavior of the air-fuel ratiomanipulating system.

[0088] The air-fuel ratio manipulating system actually includes aresponse delay element caused by the internal combustion engine 1, otherthan a dead time element. Since a response delay of theupstream-of-catalyst air-fuel-ratio with respect to the target air-fuelratio KCMD is basically compensated for by the feedback controller 14(particularly the adaptive controller 18) of the engine-side controlunit 7 b, there will arise no problem if a response delay element causedby the internal combustion engine 1 is not taken into account in theair-fuel ratio manipulating system as viewed from the exhaust-sidecontrol unit 7 a.

[0089] The target air-fuel ratio generation processor 13 according tothe present invention carries out the process for sequentiallycalculating the target air-fuel ratio KCMD according to an algorithmthat is constructed based on the exhaust system model expressed by theequation (1) and the air-fuel ratio manipulating system model expressedby the equation (2) in control cycles of the exhaust-side control unit 7a. In order to carry out the above process, the target air-fuel ratiogeneration processor 13 has its functions as shown in FIG. 3.

[0090] The target air-fuel ratio generation processor 13 comprises aflow rate data generating means 28 for sequentially calculating anestimated value ABSV of the flow rate of the exhaust gas supplied to thecatalytic converter 3 (hereinafter referred to as “estimated exhaust gasvolume ABSV”) from the detected values of the rotational speed NE andthe intake pressure PB of the internal combustion engine 1, and a deadtime setting means 29 for sequentially setting the set dead times d1, d2of the exhaust system model and the air-fuel ratio manipulating systemmodel, respectively, depending on the estimated exhaust gas volume ABSV.

[0091] Since the flow rate of the exhaust gas supplied to the catalyticconverter 3 is proportional to the product of the rotational speed NEand the intake pressure PB of the internal combustion engine 1, the deadtime setting means 29 sequentially calculates the estimated exhaust gasvolume ABSV from the detected values (present values) of the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1according to the following equation (3): $\begin{matrix}{{ABSV} = {\frac{NE}{1500} \cdot {PB} \cdot {SVPRA}}} & (3)\end{matrix}$

[0092] In the equation (3), SVPRA represents a predetermined constantdepending on the displacement (cylinder volume) of the internalcombustion engine 1. In the present embodiment, the flow rate of theexhaust gas when the rotational speed NE of the internal combustionengine 1 is 1500 rpm is used as a reference. Therefore, the rotationalspeed NE is divided by “1500” in the above equation (3).

[0093] The dead time setting means 29 sequentially determines the setdead time d1 as a value representing the actual dead time of the exhaustsystem E from the value of the estimated gas volume ABSV sequentiallycalculated by the flow rate data generating means 28 according to a datatable that is preset as indicated by the solid-line curve c in FIG. 4,for example. Similarly, the dead time setting means 29 sequentiallydetermines the set dead time d2 as a value representing the actual deadtime of the air-fuel ratio manipulating system from the value of theestimated gas volume ABSV according to a data table that is preset asindicated by the solid-line curve d in FIG. 4.

[0094] The above data tables are established based on experimentation orsimulation. Since the actual dead time of the exhaust system E isbasically longer as the flow rate of the exhaust gas supplied to thecatalytic converter 3 is smaller, as described above, the set dead timed1 represented by the solid-line curve c in FIG. 4 varies according tosuch a tendency with respect to the estimated gas volume ABSV. Likewise,since the actual dead time of the air-fuel ratio manipulating system isbasically longer as the flow rate of the exhaust gas supplied to thecatalytic converter 3 is smaller, the set dead time d2 represented bythe solid-line curve d in FIG. 4 varies according to such a tendencywith respect to the estimated gas volume ABSV. Moreover, inasmuch as thedegree of changes of the actual dead time of the air-fuel ratiomanipulating system with respect to the flow rate of the exhaust gas issmaller than the degree of changes of the actual dead time of theexhaust system E, the degree of changes of the set dead time d2 withrespect to the estimated gas volume ABSV is smaller than the degree ofchanges of the set dead time d1 in the data table shown in FIG. 4.

[0095] In the data table shown in FIG. 4, the set dead times d1, d2continuously change with respect to the estimated gas volume ABSV. Sincethe set dead times d1, d2 in the exhaust system model and the air-fuelratio manipulating system model are expressed in terms of the number ofcontrol cycles of the exhaust-side control unit 7 a, the set dead timesd1, d2 need to be of integral values. Therefore, the dead time settingmeans 29 actually determines, as set dead times d1, d2, values that areproduced by rounding off the fractions of the values of the set deadtimes d1, d2 that are determined based on the data table shown in FIG.4, for example.

[0096] In the present embodiment, the flow rate of the exhaust gassupplied to the catalytic converter 3 is estimated from the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1.However, the flow rate of the exhaust gas may be directly determinedusing a flow sensor or the like.

[0097] The target air-fuel ratio generation processor 13 comprises anidentifier (identifying means) 25 for sequentially identifying values ofthe gain coefficients a1, a2, b1 that are parameters for the exhaustsystem model, an estimator (estimating means) 26 for sequentiallydetermining in each control cycle an estimated value VO2 bar of thedifferential output VO2 from the O₂ sensor 6 (hereinafter referred to as“estimated differential output VO2 bar”) after the total set dead time d(=d1+d2) which is the sum of the set dead time d1 of the exhaust systemE and the set dead time d2 of the air-fuel ratio manipulating system,and a sliding mode controller 27 for sequentially determining the targetair-fuel ratio KCMD according to an adaptive sliding mode controlprocess.

[0098] The algorithm of a processing operation to be carried out by theidentifier 25, the estimator 26, and the sliding mode controller 27 isconstructed based on the exhaust system model and the air-fuel ratiomanipulating system model, as follows:

[0099] With respect to the identifier 25, the gain coefficients of theactual exhaust system E which correspond to the gain coefficients a1,a2, b1 of the exhaust system model generally change depending on thebehavior of the exhaust system E and chronological characteristicchanges of the exhaust system E. Therefore, in order to minimize amodeling error of the exhaust system model (the equation (1)) withrespect to the actual exhaust system E for increasing the accuracy ofthe model, it is preferable to identify the gain coefficients a1, a2, b1in real-time suitably depending on the actual behavior of the exhaustsystem E.

[0100] The identifier 25 serves to identify the gain coefficients a1,a2, b1 sequentially on a real-time basis for the purpose of minimizing amodeling error of the exhaust system model. The identifier 25 carriesout its identifying process as follows:

[0101] In each control cycle of the exhaust-side control unit 7 a, theidentifier 25 determines an identified value VO2(k) hat of thedifferential output VO2 from the O₂ sensor 6 (hereinafter referred to as“identified differential output VO2(k) hat”) on the exhaust systemmodel, using the identified gain coefficients a1 hat, a2 hat, b1 hat ofthe presently established exhaust system model, i.e., identified gaincoefficients a1(k−1) hat, a2(k−1) hat, b1(k−1) hat determined in apreceding control cycle, past data of the differential output kact fromthe LAF sensor 5 and the differential output VO2 from the O₂ sensor 6,and the latest value of the set dead time d1 of the exhaust system Ethat has been set by the dead time setting means 29, according to thefollowing equation (4):

VÔ 2(k)=a{circumflex over (1)}(k−1)·VO 2(k−1) +a{circumflex over(2)}(k−1)·VO 2(k−2)+b{circumflex over (1)}(k−1)·kact(k−d 1−1)  (4)

[0102] The equation (4) corresponds to the equation (1) which is shiftedinto the past by one control cycle with the gain coefficients a1, a2, b1being replaced with the respective identified gain coefficients a1(k−1)hat, a2(k−1) hat, b1(k−1) hat, and the latest value of the set dead timed1 used as the dead time d1 of the exhaust system E.

[0103] If vectors Θ, ξ defined by the following equations (5), (6) areintroduced (the letter T in the equations (5), (6) represents atransposition), then the equation (4) is expressed by the equation (7):

Θ^(T)(k)=[â1(k)â2(k){circumflex over (b)}1(k)]  (5)

ξ^(T)(k)=[VO 2(k−1)VO 2(k−2) kact(k−d 1−1)]  (6)

VÔ2(k)=Θ^(T)(k−1)·ξ(k)  (7)

[0104] The identifier 25 also determines a difference id/e(k) betweenthe identified differential output VO2(k) hat from the O₂ sensor 6 whichis determined by the equation (4) or (7) and the present differentialoutput VO2(k) from the O₂ sensor 6, as representing a modeling error ofthe exhaust system model with respect to the actual exhaust system E(hereinafter the difference id/e will be referred to as “identifiederror id/e”), according to the following equation (8):

id/e(k)=VO 2(k)−VÔ2(k)  (8)

[0105] The identifier 25 further determines new identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat, stated otherwise, a newvector Θ (k) having these identified gain coefficients as elements(hereinafter the new vector Θ (k) will be referred to as “identifiedgain coefficient vector Θ”), in order to minimize the identified errorid/e, according to the equation (9) given below. That is, the identifier25 varies the identified gain coefficients a1 hat (k−1), a2 hat (k−1),b1 hat,(k−1) determined in the preceding control cycle by a quantityproportional to the identified error id/e for thereby determining thenew identified gain coefficients a1(k) hat, a2(k) hat, b1(k) hat.

Θ(k)=Θ(k−1)+Kθ(k)·id/e(k)  (9)

[0106] where Kθ represents a cubic vector determined by the followingequation (10) (a gain coefficient vector for determining a changedepending on the identified error id/e of each of the identified gaincoefficients a1 hat, a2 hat, b1 hat): $\begin{matrix}{{K\quad {\theta (k)}} = \frac{{P\left( {k - 1} \right)} \cdot {\xi (k)}}{1 + {{\xi^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)}}}} & (10)\end{matrix}$

[0107] where P represents a cubic square matrix determined by arecursive formula expressed by the following equation (11):$\begin{matrix}{{{P(k)} = {\frac{1}{\lambda_{1}(k)} \cdot \left\lbrack {I - \frac{{\lambda_{2}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)} \cdot {\xi^{T}(k)}}{{\lambda_{1}(k)} + {{\lambda_{2}(k)} \cdot {\xi^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi (k)}}}} \right\rbrack \cdot {P\left( {k - 1} \right)}}}{{where}\quad I\quad {represents}\quad a\quad {unit}\quad {{matrix}.}}} & (11)\end{matrix}$

[0108] In the equation (11), λ₁, λ₂ are established to satisfy theconditions 0<λ₁ ≦1 and 0≦λ₂<2, and an initial value P(0) of P representsa diagonal matrix whose diagonal components are positive numbers.

[0109] Depending on how the values of λ₁, λ₂ in the equation (11) areestablished, any one of various specific algorithms including a fixedgain method, a degressive gain method, a method of weighted leastsquares, a method of least squares, a fixed tracing method, etc. may beemployed. According to the present embodiment, the algorithm of a methodof weighted least squares is employed, and the values of λ₁, λ₂ are that0<λ₁, <1, λ₂=1.

[0110] “λ₁” represents a weighted parameter according to a method ofweighted least squares. In the present embodiment, the value of theweighted parameter λ₁ is variably set depending on the estimated exhaustgas volume ABSV that is sequentially calculated by the flow rate datagenerating means 28 (as a result, depending on the set dead time d1).

[0111] Specifically, in the present embodiment, the identifier 25 sets,in each control cycle of the exhaust-side control unit 7 a, the value ofthe weighted parameter λ₁ from the latest value of the estimated exhaustgas volume ABSV determined by the flow rate data generating means 28,based on a predetermined data table shown in FIG. 5. In the data tableshown in FIG. 5, the value of the weighted parameter λ₁ is greater,approaching “1”, as the estimated exhaust gas volume ABSV is smaller.The identifier 25 uses the value of the weighted parameter λ₁ thus setdepending on the estimated exhaust gas volume ABSV for updating thematrix P(k) according to the equation (11) in each control cycle.

[0112] Basically, the identifier 25 sequentially determines in eachcontrol cycle the identified gain coefficients a1 hat, a2 hat, b1 hat ofthe exhaust system model according to the above algorithm (calculatingoperation), i.e., the algorithm of a sequential method of weighted leastsquares, in order to minimize the identified error id/e.

[0113] The calculating operation described above is the basic algorithmthat is carried out by the identifier 25. The identifier 25 performsadditional processes such as a limiting process, on the identified gaincoefficients a1 hat, a2 hat, b1 hat in order to determine them. Suchoperations of the identifier 25 will be described later on.

[0114] The estimator 26 sequentially determines in each control cyclethe estimated differential output VO2 bar which is an estimated value ofthe differential output VO2 from the O₂ sensor 6 after the total setdead time d (=d1+d2) in order to compensate for the effect of the deadtime d1 of the exhaust system E and the effect of the dead time d2 ofthe air-fuel ratio manipulating system for the calculation of the targetair-fuel ratio KCMD with the sliding mode controller 27 as described indetail later on. The algorithm for the estimator 26 to determine theestimated differential output VO2 bar is constructed as follows:

[0115] If the equation (2) expressing the air-fuel ratio manipulatingsystem model is applied to the equation (1) expressing the exhaustsystem model, then the equation (1) can be rewritten as the followingequation (12): $\begin{matrix}{{{VO2}\left( {k + 1} \right)} = {{{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} + {{b1} \cdot {{kcmd}\left( {k - {d1} - {d2}} \right)}}}\quad = {{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} + {{b1} \cdot {{kcmd}\left( {k - d} \right)}}}}} & (12)\end{matrix}$

[0116] The equation (12) expresses the behavior of a system which is acombination of the exhaust system E and the air-fuel manipulating systemas a discrete time system, regarding such a system as a system forgenerating the differential output VO2 from the O₂ sensor 6 from thetarget differential air-fuel ratio kcmd via dead time elements of theexhaust system E and the air-fuel manipulating system and a responsedelay element of the exhaust system E.

[0117] By using the equation (12), the estimated differential outputVO2(k+d) bar after the total set dead time d in each control cycle canbe expressed using time-series data VO2(k), VO2(k−1) of present and pastvalues of the differential output VO2 of the O₂ sensor 6 and time-seriesdata kcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd (=KCMD−FLAF/BASE) which corresponds tothe target air-fuel ratio KCMD determined by the sliding mode controller27 (its specific process of determining the target air-fuel ratio KCMDwill be described later on), according to the following equation (13):$\begin{matrix}{{{\overset{\_}{VO2}\left( {k + d} \right)} = {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{d}\quad {\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}}}}{where}{{{\alpha 1} = {{the}\quad {first}\text{-}{row}}},{{first}\text{-}{column}\quad {element}\quad {of}\quad A^{d}},{{\alpha 2} = {{the}\quad {first}\text{-}{row}}},{{second}\text{-}{column}\quad {element}\quad {of}\quad A^{d}},{{\beta \quad j} = {{the}\quad {first}\text{-}{row}\quad {elements}\quad {of}\quad {A^{j - 1} \cdot B}}}}{A = \begin{bmatrix}{a1} & {a2} \\1 & 0\end{bmatrix}}{B = \begin{bmatrix}{b1} \\0\end{bmatrix}}} & (13)\end{matrix}$

[0118] In the equation (13), “α1”, “α2” represent the first-row,first-column element and the first-row, second-column element,respectively, of the power A^(d) (d: total dead time) of the matrix Adefined as described above with respect to the equation (13), and “βj”(j=1, 2, . . . , d) represents the first-row elements of the productA^(j−1)·B of the power A^(j−1) (j=1, 2, . . . , d) of the matrix A andthe vector B defined as described above with respect to the equation(13).

[0119] Of the time-series data kcmd(k−j) (j=1, 2, . . . , d) of the pastvalues of the target combined differential air-fuel ratio kcmd accordingto the equation (13), the time-series data kcmd(k−d2), kcmd(k−d2−1), . .. , kcmd(k−d) from the present prior to the dead time d2 of the air-fuelmanipulating system can be replaced respectively with data kact(k),kact(k−1), . . . , kact(k−d+d2) obtained prior to the present time ofthe differential output kact of the LAF sensor 5 according the aboveequation (2). When the time-series data are thus replaced, the followingequation (14) is obtained: $\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{{d2} - 1}\quad {\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}} + {\sum\limits_{i = 0}^{d - {d2}}\quad {\beta \quad i}} + {{d2} \cdot {{kact}\left( {k - i} \right)}}} = {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{{d2} - 1}\quad {\beta \quad {j \cdot {{kcmd}\left( {k - j} \right)}}}} + {\sum\limits_{i = 0}^{d1}\quad {\beta \quad i}} + {{d2} \cdot {{kact}\left( {k - i} \right)}}}}} & (14)\end{matrix}$

[0120] The equation (14) is a basic formula for the estimator 26 tosequentially determine the estimated differential output VO2(k+d) bar.Stated otherwise, the estimator 26 determines, in each control cycle,the estimated differential output VO2(k+d) bar of the O₂ sensor 6according to the equation (14), using the time-series data VO2(k),VO2(k−1) of the present and past values of the differential output VO2of the O₂ sensor 6, the time-series data kcmd(k−j) (j =1, . . . , d2−1)of the past values of the target differential air-fuel ratio kcmd whichrepresents the target air-fuel ratio KCMD determined in the past by thesliding mode controller 27, and the time-series data kact(k−i) (i=0, . .. , d1) of the present and past values of the differential output kactof the LAF sensor 5.

[0121] The values of the coefficients α1, α2, βj (j=1, 2, . . . , d)required to calculate the estimated differential output VO2(k+d) baraccording to the equation (14) basically employ the identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat which are the latestidentified values of the gain coefficients a1, a2, b1 (which areelements of the matrix A and vector B defined with respect to theequation (13)). The values of the dead times d1, d2 required in theequation (14) comprise the latest values of the set dead times d1, d2that are set by the dead time setting means 29 as described above.

[0122] In the present embodiment, the set dead times d1, d2 used in theequation (14) change depending on the estimated exhaust gas volume ABSV,and the number of data of the target differential air-fuel ratio kcmdand data of the differential output kact of the LAF sensor 5 which arerequired to calculate the estimated differential output VO(k+d) baraccording to the equation (14) also changes depending on the set deadtimes d1, d2. In this case, the set dead time d2 of the air-fuel ratiomanipulating system may become “1” (in the present embodiment d1>d2≧1,see FIG. 4). If the dead time d2 of the air-fuel ratio manipu-latingsystem becomes “1”, then all the time-series data kcmd(k−j) (j=1, 2, . .. , d) of the past values of the target differential air-fuel ratio kcmdin the equation (13) may be replaced with the time-series data kact(k),kact(k−1), . . . , kact(k−d+d2), respectively, prior to the presenttime, of the differential output kact of the LAF sensor 5. In this case,the equation (13) is rewritten into the following equation (15) whichdoes not include the data of the target differential air-fuel ratiokcmd: $\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 0}^{d - 1}\quad {\beta \quad j}} + {1 \cdot {{kact}\left( {k - j} \right)}}}} & (15)\end{matrix}$

[0123] Specifically, if the value of the set dead time d2 is “1”, thenthe estimated differential output VO2(k+d) bar of the O₂ sensor 6 can bedetermined using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6, the time-series datakact(k−j) (j=0, 1, . . . , d−1) of the present and past values of thedifferential output kact of the LAF sensor 5, the coefficients α1, α2,βj (j=1, 2, . . . , d) determined by the identified gain coefficients a1hat, a2 hat, b1 hat, and the total set dead time d (=d1+d2) which is thesum of the set dead times d1, d2.

[0124] In the present embodiment, therefore, if the set dead time d2 isd2>1, then the estimator 26 determines the estimated differential outputVO2(k+d) bar according to the equation (14), and if the set dead time d2is d2=1, then the estimator 26 determines the estimated differentialoutput VO2(k+d) bar according to the equation (15).

[0125] The estimated differential output VO2(k+d) bar may be determinedaccording to the equation (13) without using the data of thedifferential output kact of the LAF sensor 5. In this case, theestimated differential output VO2(k+d) bar of the O₂ sensor 6 isdetermined using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6, the time-series datakcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd, the coefficients α1, α2, βj (j=1, 2, .. . , d) determined by the identified gain coefficients a1 hat, a2 hat,b1 hat, and the total set dead time d (=d1+d2) which is the sum of theset dead times d1, d2. It is also possible to determine the estimateddifferential output VO2(k+d) bar according to an equation where only aportion of the time-series data of the target differential air-fuelratio kcmd prior to the set dead time d2 in the equation (13) isreplaced with the differential output kact of the LAF sensor 5. However,for increasing the reliability of the estimated differential outputVO2(k+d) bar, it is preferable to determine the estimated differentialoutput VO2(k+d) bar according to the equation (14) or (15) which uses,as much as possible, the data of the differential output kact of the LAFsensor 5 that reflects the actual behavior of the internal combustionengine 1, etc.

[0126] The algorithm described above is a basic algorithm for theestimator 26 to determine, in each control cycle, the estimateddifferential output VO2(k+d) bar that is an estimated value after thetotal dead time d of the differential output VO2 of the O₂ sensor 6.

[0127] The sliding mode controller 27 will be described in detail below.

[0128] The sliding mode controller 27 sequentially calculates an inputquantity to be given to the exhaust system E (which is specifically atarget value for the difference between the output KACT of the LAFsensor 5 (the detected value of the air-fuel ratio) and the air-fuelratio reference value FLAF/BASE, which is equal to the targetdifferential air-fuel ratio kcmd, the input quantity will be referred toas “SLD manipulating input Usl”) in order to cause the output VO2/OUT ofthe O₂ sensor 6 to converge to the target value VO2/TARGET (to convergethe differential output VO2 of the O₂ sensor 6 to “0”) according to anadaptive sliding mode control process which incorporates an adaptivecontrol law (adaptive algorithm) for minimizing the effect of adisturbance, in a normal sliding mode control process, and sequentiallydetermines the target air-fuel ratio KCMD from the calculated SLDmanipulating input Usl. An algorithm for carrying out the adaptivesliding mode control process is constructed as follows:

[0129] A switching function required for the adaptive sliding modecontrol process carried out by the sliding mode controller 27 and ahyperplane defined by the switching function (also referred to as a slipplane) will first be described below.

[0130] According to a basic concept of the sliding mode control processin the present embodiment, the differential output VO2(k) of the O₂sensor 6 obtained in each control cycle and the differential outputVO2(k−1) obtained in a preceding control cycle are used as a statequantity to be controlled, and a switching function σ for the slidingmode control process is defined according to the equation (16) shownbelow. Specifically, the switching function σ is defined by a linearfunction whose components are represented by the time-series dataVO2(k), VO2(k−1) of the differential output VO2 of the O₂ sensor 6. Thevector X defined in equation 16 below as the vector having thedifferential output VO2(k), VO2(k−1) as elements thereof is hereinafterreferred to as “state quantity X”. $\begin{matrix}{{{\sigma (k)} = {{{{s1} \cdot {{VO2}(k)}} + {{s2} \cdot {{VO2}\left( {k - 1} \right)}}} = {S \cdot X}}}{where}{{S = \left\lbrack {{s1}\quad {s2}} \right\rbrack},{X = \begin{bmatrix}{{VO2}(k)} \\{{VO2}\left( {k - 1} \right)}\end{bmatrix}}}} & (16)\end{matrix}$

[0131] The coefficients s1, s2 relative to the respective componentsVO2(k), VO2(k−1) of the switching function σ are set in order to meetthe condition of the following equation (17): $\begin{matrix}{{{- 1} < \frac{s2}{s1} < 1}\left( {{{{when}\quad {s1}} = 1},{{- 1} < {s2} < 1}} \right)} & (17)\end{matrix}$

[0132] In the present embodiment, for the sake of brevity, thecoefficient s1 is set to s1=1 (s2/s1=s2), and the value of thecoefficient s2 is established to satisfy the condition: −1<s2<1.

[0133] With the switching function σ thus defined, the hyperplane forthe sliding mode control process is defined by the equation σ=0. Sincethe state quantity X is of the second degree, the hyperplane σ=0 isrepresented by a straight line as shown in FIG. 6. The hyperplane iscalled a switching line or a switching plane depending on the degree ofa topological space.

[0134] In the present embodiment, the time-series data of the estimateddifferential output VO2 bar determined by the estimator 26 is used as astate quantity representative of the variable components of theswitching function, as described later on.

[0135] The adaptive sliding mode control process used in the presentembodiment serves to converge the state quantity X onto the hyperplaneσ=0 according to a reaching law which is a control law for convergingthe state quantity X (=VO2(k), VO2(k−1)) onto the hyperplane σ=0 (forconverging the value of the switching function σ to “0”) and an adaptivelaw (adaptive algorithm) which is a control law for compensating for theeffect of a disturbance in converging the state quantity X onto thehyperplane σ=0 (mode 1 in FIG. 6). While holding the state quantity Xonto the hyper-plane σ=0 according to a so-called equivalent controlinput, the state quantity X is converged to a balanced point on thehyperplane σ=0 where VO2(k)=VO2(k−1)=0, i.e., a point where time-seriesdata VO2/OUT(k), VO2/OUT(k−1) of the output VO2/OUT of the O₂ sensor 6are equal to the target value VO2/TARGET (mode 2 in FIG. 6).

[0136] The SLD manipulating input Usl (=the target differential air-fuelratio kcmd) to be generated by the sliding mode controller 27 forconverging the state quantity X toward the balanced point on thehyperplane σ=0 is expressed as the sum of an equivalent control inputUeq to be applied to the exhaust system E according to the control lawfor converging the state quantity X onto the hyperplane σ=0, an inputquantity component Urch (hereinafter referred to as “reaching law inputUrch”) to be applied to the exhaust system E according to the reachinglaw, and an input quantity component Uadp (hereinafter referred to as“adaptive law input Uadp”) to be applied to the exhaust system Eaccording to the adaptive law, according to the following equation (18).

Usl=Ueq+Urch+Uadp  (18)

[0137] In the present embodiment, the equivalent control input Ueq, thereaching law input Urch, and the adaptive law input Uadp are determinedon the basis of the above equation (12) where the exhaust system modeland the air-fuel ratio manipulating system model are combined, asfollows:

[0138] The equivalent control input Ueq which is an input quantitycomponent to be applied to the exhaust system E for holding the statequantity X on the hyperplane σ=0 is the target differential air-fuelratio kcmd which satisfies the condition: σ(k+1)=σ(k)=0. Using theequations (12), (16), the equivalent control input Ueq which satisfiesthe above condition is given by the following equation (19):$\begin{matrix}{{{Ueq}(k)} = {{{- \left( {S \cdot B} \right)^{- 1}} \cdot \left\{ {S \cdot \left( {A - 1} \right)} \right\} \cdot {X\left( {k + d} \right)}}\quad = {\frac{- 1}{s1b1} \cdot \left\{ {{{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot V}\quad {{O2}\left( {k + d} \right)}} + {{\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot V}\quad {{O2}\left( {k + d - 1} \right)}}} \right\}}}} & (19)\end{matrix}$

[0139] The equation (19) is a basic formula for determining theequivalent control input Ueq(k) in each control cycle.

[0140] According to the present embodiment, the reaching law input Urchis basically determined according to the following equation (20):$\begin{matrix}\begin{matrix}{{{Urch}(k)} = {{- \left( {S \cdot B} \right)^{- 1}} \cdot F \cdot {\sigma \left( {k + d} \right)}}} \\{= {\frac{- 1}{s1b1} \cdot F \cdot {\sigma \left( {k + d} \right)}}}\end{matrix} & (20)\end{matrix}$

[0141] Specifically, the reaching law input Urch is determined inproportion to the value σ(k+d) of the switching function a after thetotal dead time d, in view of the dead times of the exhaust system E andthe air-fuel ratio manipu- lating system.

[0142] The coefficient F in the equation (20) (which determines the gainof the reaching law) is established to satisfy the condition expressedby the following equation (21):

0<F<2 (preferably, 0<F<1)  (21)

[0143] The condition of the equation (21) is a condition for stablyconverging the value of the switching function a onto the hyperplaneσ=0. The preferable condition in the equation (21) is a conditionsuitable for preventing the value of the switching function a fromoscillating (so-called chattering) with respect to the hyperplane σ=0.

[0144] In the present embodiment, the adaptive law input Uadp isbasically determined according to the following equation (22) (ΔT in theequation (22) represents the period of the control cycles of theexhaust-side control unit 7 a): $\begin{matrix}\begin{matrix}{{{Uadp}(k)} = {{- \left( {S \cdot B} \right)^{- 1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma (i)} \cdot \Delta}\quad T} \right)}}} \\{= {\frac{- 1}{s1b1} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma (i)} \cdot \Delta}\quad T} \right)}}}\end{matrix} & (22)\end{matrix}$

[0145] The adaptive law input Uadp is determined in proportion to anintegrated value (which corresponds to an integral of the values of theswitching function σ) of the product of values of the switching functionσ and the period ΔT of the control cycles of the exhaust-side controlunit 7 a until after the total dead time d, in view of the dead times ofthe exhaust system E and the air-fuel ratio manipulating system.

[0146] The coefficient G (which determines the gain of the adaptive law)in the equation (22) is established to satisfy the condition of thefollowing equation (23): $\begin{matrix}{{G = {J \cdot \frac{2 - F}{\Delta \quad T}}}\left( {0 < J < 2} \right)} & (23)\end{matrix}$

[0147] The condition of the equation (23) is a condition for convergingthe value of the switching function a stably onto the hyperplane σ=0regardless of disturbances, etc.

[0148] A specific process of deriving conditions for establishing theequations (17), (21), (23) is described in detail in Japanese patentapplication No. 11-93741 by the applicant of the present application,and will not be described in detail below.

[0149] In the present embodiment, the sliding mode controller 27determines the sum (Ueq+Urch+Uadp) of the equivalent control input Ueq,the reaching law input Urch, and the adaptive law input Uadp determinedaccording to the respective equations (19), (20), (22) as the SLDmanipulating input Usl to be applied to the exhaust system E. However,the differential outputs VO2(K+d), VO2(k+d−1) of the O₂ sensor 6 and thevalue σ(k+d) of the switching function σ, etc. used in the equations(19), (20), (22) cannot directly be obtained as they are values in thefuture.

[0150] According to the present embodiment, therefore, the sliding modecontroller 27 actually uses the estimated differential outputs VO2(k+d)bar, VO2(k+d−1) bar determined by the estimator 26, instead of thedifferential outputs VO2(K+d), VO2(k+d−1) from the O₂ sensor 6 fordetermining the equivalent control input Ueq according to the equation(19), and calculates the equivalent control input Ueq in each controlcycle according to the following equation (24): $\begin{matrix}\begin{matrix}{{{Ueq}(k)} = {\frac{- 1}{s1b1}\left\{ {{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot {\overset{\_}{VO2}\left( {k + d} \right)}} +} \right.}} \\\left. {\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot {\overset{\_}{VO2}\left( {k + d - 1} \right)}} \right\}\end{matrix} & (24)\end{matrix}$

[0151] According to the present embodiment, furthermore, the slidingmode controller 27 actually uses time-series data of the estimateddifferential output VO2 bar sequentially determined by the estimator 26as described above as a state quantity to be controlled, and defines aswitching function σ bar according to the following equation (25) (theswitching function σ bar corresponds to time-series data of thedifferential output VO2 in the equation (16) which is replaced withtime-series data of the estimated differential output VO2 bar), in placeof the switching function a established according to the equation (16):

{overscore (σ(k))}=s 1·VO 2(k)+s 2·VO 2(k−1)  (25)

[0152] The sliding mode controller 27 calculates the reaching law inputUrch in each control cycle according to the following equation (26),using the switching function σ bar represented by the equation (25),rather than the value of the switching function a for determining thereaching law input Urch according to the equation (20): $\begin{matrix}{{{Urch}(k)} = {\frac{- 1}{{s1} \cdot {b1}} \cdot F \cdot {\overset{\_}{\sigma}\left( {k + d} \right)}}} & (26)\end{matrix}$

[0153] Similarly, the sliding mode controller 27 calculates the adaptivelaw input Uadp in each control cycle according to the following equation(27), using the value of the switching function a bar represented by theequation (25), rather than the value of the switching function α fordetermining the adaptive law input Uadp according to the equation (22):$\begin{matrix}{{{Uadp}(k)} = {\frac{- 1}{{s1} \cdot {b1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\overset{\_}{\sigma}(i)} \cdot \Delta}\quad T} \right)}}} & (27)\end{matrix}$

[0154] The latest identified gain coefficients a1(k) hat, a2(k) hat,b1(k) hat which have been determined by the identifier 25 are basicallyused as the gain coefficients a1, a1, b1 that are required to calculatethe equivalent control input Ueq, the reaching law input Urch, and theadaptive law input Uadp according to the equations (24), (26), (27). Thevalues of the switching function σ bar in each control cycle which arerequired to calculate the reaching law input Urch and the adaptive lawinput Uadp are represented by the latest estimated differential outputVO2(k+1) bar determined by the estimator 26 and the estimateddifferential output VO2(k+d−1) bar determined by the estimator 26 in thepreceding control cycle.

[0155] The sliding mode controller 27 determines the sum of theequivalent control input Ueq, the reaching law input Urch, and theadaptive law input Uadp determined according to the equations (24),(26), (27), as the SLD manipulating input Usl to be applied to theexhaust system E (see the equation (18)). The conditions forestablishing the coefficients s1, s2, F, G used in the equations (24),(26), (27) are as described above.

[0156] The above process is a basic algorithm for the sliding modecontroller 27 to determine the SLD manipulating input Usl (=targetdifferential air-fuel ratio kcmd) to be applied to the exhaust system E.According to the above algorithm, the SLD manipulating input Usl isdetermined to converge the estimated differential output VO2 bar fromthe O₂ sensor 6 to “0” (as a result, to converge the output VO2/OUT fromthe O₂ sensor 6 to the target value VO2/TARGET).

[0157] The sliding mode controller 27 eventually sequentially determinesthe target air-fuel ratio KCMD in each control cycle. The SLDmanipulating input Usl determined as described above signifies a targetvalue for the difference between the upstream-of-catalyst air-fuel ratiodetected by the LAF sensor 5 and the air-fuel ratio reference valueFLAF/BASE, i.e., the target differential air-fuel ratio kcmd.Consequently, the sliding mode controller 27 eventually determines thetarget air-fuel ratio KCMD by adding the reference value FLAF/BASE tothe determined SLD manipulating input Usl in each control cycleaccording to the following equation (28): $\begin{matrix}\begin{matrix}{{{KCMD}(k)} = {{{Usl}(k)} + {{FLAF}/{BASE}}}} \\{= {{{Ueq}(k)} + {{Urch}(k)} + {{Uadp}(k)} + {{FLAF}/{BASE}}}}\end{matrix} & (28)\end{matrix}$

[0158] The above process is a basic algorithm for the sliding modecontroller 27 to sequentially determine the target air-fuel ratio KCMDaccording to the present embodiment.

[0159] In the present embodiment, the stability of the adaptive slidingmode control process carried out by the sliding mode controller 27 ischecked for limiting the value of the SLD manipulating input Usl.Details of such a checking process will be described later on.

[0160] The general feedback controller 15 of the engine-side controlunit 7 b, particularly, the adaptive controller 18, will further bedescribed below.

[0161] In FIG. 1, the general feedback controller 15 effects a feedbackcontrol process to converge the output KACT from the LAF sensor 5 to thetarget air-fuel ratio KCMD as described above. If such a feedbackcontrol process were carried out under the known PID control only, itwould be difficult to keep stable controllability against dynamicbehavioral changes including changes in the operating state of theinternal combustion engine 1, characteristic changes due to aging of theinternal combustion engine 1, etc.

[0162] The adaptive controller 18 is a recursive-type controller whichmakes it possible to carry out a feedback control process whilecompensating for dynamic behavioral changes of the internal combustionengine 1. As shown in FIG. 7, the adaptive controller 18 comprises aparameter adjuster 30 for establishing a plurality of adaptiveparameters using the parameter adjusting law proposed by I. D. Landau,et al., and a manipulated variable calculator 31 for calculating thefeedback manipulated variable KSTR using the established adaptiveparameters.

[0163] The parameter adjuster 30 will be described below. According tothe adjusting law proposed by I. D. Landau, et al., when polynomials ofthe denominator and the numerator of a transfer function B(Z⁻¹)/A(Z⁻¹)of a discrete-system object to be controlled are generally expressedrespectively by equations (29), (30), given below, an adaptive parameterθ hat (j) (j indicates the ordinal number of a control cycle of theengine-side control unit 7 b) established by the parameter adjuster 30is represented by a vector (transposed vector) according to the equation(31) given below. An input ζ (j) to the parameter adjuster 30 isexpressed by the equation (32) given below. In the present embodiment,it is assumed that the internal combustion engine 1, which is an objectto be controlled by the general feedback controller 15, is considered tobe a plant of a first-order system having a dead time dp (the time ofthree combustion cycles of the internal combustion engine 1), and m=n=1,dp=3 in the equations (29)-(32), and five adaptive parameters s0, r1,r2, r3, b0 are established (see FIG. 7). In the upper and middleexpressions of the equation (34), us, ys generally represent an input(manipulated variable) to the object to be controlled and an output(controlled variable) from the object to be controlled. In the presentembodiment, the input is the feedback manipulated variable KSTR and theoutput from the object to be controlled (the internal combustion engine1) is the output KACT (upstream-of-catalyst air-fuel ratio) from the LAFsensor 5, and the input ζ (j) to the parameter adjuster 30 is expressedby the lower expression of the equation (32) (see FIG. 7).

A(Z ⁻¹)=1+a 1 Z ⁻¹ +. . . +anZ ^(−n)  (29)

B(Z ⁻¹)=b 0+b 1 Z ⁻¹ +. . . +bmZ ^(−m)  (30)

[0164] $\begin{matrix}\begin{matrix}{{{\hat{\theta}}^{T}(j)} = \left\lbrack {{\hat{b}0(j)},{\hat{B}{R\left( {Z^{- 1},j} \right)}},{\hat{S}\left( {Z^{- 1},j} \right)}} \right\rbrack} \\{= \left\lbrack {{{b0}(j)},{{r1}(j)},\cdots \quad,{{rm} +}} \right.} \\\left. {{{dp} - {1(j)}},{{s0}(j)},\cdots \quad,{{sn} - {1(j)}}} \right\rbrack \\{= \left\lbrack {{{b0}(j)},{{r1}(j)},{{r2}(j)},{{r3}(j)},{{s0}(j)}} \right\rbrack}\end{matrix} & (31) \\\begin{matrix}{{\zeta^{T}(j)} = \left\lbrack {{{us}(j)},\cdots \quad,{{us}\left( {j - m - {dp} + 1} \right)},{{ys}(j)},\cdots \quad,} \right.} \\\left. {{ys}\left( {j - n + 1} \right)} \right\rbrack \\{= \left\lbrack {{{us}(j)},{{us}\left( {j - 1} \right)},{{us}\left( {j - 2} \right)},{{us}\left( {j - 3} \right)},{{ys}(j)}} \right\rbrack} \\{= \left\lbrack {{{KSTR}(j)},{{KSTR}\left( {j - 1} \right)},{{KSTR}\left( {j - 2} \right)},{{KSTR}\left( {j - 3} \right)},} \right.} \\\left. {{KACT}(j)} \right\rbrack\end{matrix} & (32)\end{matrix}$

[0165] The adaptive parameter θ hat expressed by the equation (36) ismade up of a scalar quantity element b0 hat (j) for determining the gainof the adaptive controller 18, a control element BR hat (Z⁻¹,j)expressed using a manipulated variable, and a control element S (Z⁻¹,j)expressed using a controlled variable, which are expressed respectivelyby the following equations (33) through (35) (see the block of themanipulated variable calculator 31 shown in FIG. 7): $\begin{matrix}{{\hat{b}0^{- 1}(j)} = \frac{1}{b0}} & (33) \\\begin{matrix}{{\hat{B}{R\left( {Z^{- 1},j} \right)}} = {{r1Z}^{- 1} + {z2Z}^{- 2} + \cdots + {rm} + {dp} - {1Z^{- {({n + {dp} - 1})}}}}} \\{= {{r1Z}^{- 1} + {r2Z}^{- 2} + {r3Z}^{- 3}}}\end{matrix} & (34) \\\begin{matrix}{{\hat{S}\left( {Z^{- 1},j} \right)} = {{s0} + {s1Z}^{- 1} + \quad \cdots \quad + {sn} - {1Z^{- {({n - 1})}}}}} \\{= {s0}}\end{matrix} & (35)\end{matrix}$

[0166] The parameter adjuster 30 establishes coefficients of the scalarquantity element and the control elements, described above, and suppliesthem as the adaptive parameter θ hat expressed by the equation (31) tothe manipulated variable calculator 31. The parameter adjuster 30calculates the adaptive parameter θ hat so that the output KACT from theLAF sensor 5 will agree with the target air-fuel ratio KCMD, usingtime-series data of the feedback manipulated variable KSTR from thepresent to the past and the output KACT from the LAF sensor 5.

[0167] Specifically, the parameter adjuster 30 calculates the adaptiveparameter θ hat according to the follow- ing equation (36):

{circumflex over (θ)}j)={circumflex over(θ)}(j−1)+Γ(j−1)·ζ(j−dp)·e*(j)  (36)

[0168] where Γ (j) represents a gain matrix (whose degree is indicatedby m+n+dp) for determining a rate of establishing the adaptive parameterθ hat, and e*(j) an estimated error of the adaptive parameter θ hat. Γ(j) and e*(j) are expressed respectively by the following recursiveformulas (37), (38): $\begin{matrix}{{\Gamma (j)} = {\frac{1}{{\lambda l}(j)} \cdot \left\lbrack {{\Gamma \left( {j - 1} \right)} - \frac{\lambda \quad 2{(j) \cdot {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)} \cdot \quad {\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma \left( {j - 1} \right)}}}{{{\lambda 1}(j)} + {{{\lambda 2}(j)} \cdot {\zeta^{T}\left( {j - {dp}} \right)} \cdot \quad {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}} \right\rbrack}} & (37)\end{matrix}$

[0169] where 0<λ1(j)≦1, 0 ≦λ2(j)<2, Γ (0)>0. $\begin{matrix}{{e*(j)} = \frac{{{D\left( Z^{- 1} \right)} \cdot {{KACT}(j)}} - {{{\hat{\theta}}^{T}\left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}{1 + {{\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma \left( {j - 1} \right)} \cdot {\zeta \left( {j - {dp}} \right)}}}} & (38)\end{matrix}$

[0170] where D(Z⁻¹) represents an asymptotically stable polynomial foradjusting the convergence. In the present embodiment, D(Z⁻¹)=1.

[0171] Various specific algorithms including the degressive gainalgorithm, the variable gain algorithm, the fixed tracing algorithm, andthe fixed gain algorithm are obtained depending on how λ1(j), λ2(j) inthe equation (37) are selected. For a time-dependent plant such as afuel injection process, an air-fuel ratio, or the like of the internalcombustion engine 1, either one of the degressive gain algorithm, thevariable gain algorithm, the fixed gain algorithm, and the fixed tracingalgorithm is suitable.

[0172] Using the adaptive parameter θ hat (s0, r1, r2, r3, b0)established by the parameter adjuster 30 and the target air-fuel ratioKCMD determined by the target air-fuel ratio generation processor 13,the manipulated variable calculator 31 determines the feedbackmanipulated variable KSTR according to a recursive formula expressed bythe following equation (39): $\begin{matrix}{{KSTR} = \frac{\begin{matrix}{{{KCMD}(j)} - {{S0} \cdot {{KACT}(j)}} - {{r1} \cdot {{KSTR}\left( {j - 1} \right)}} -} \\{{{r2} \cdot {{KSTR}\left( {j - 2} \right)}} - {{r3} \cdot {{KSTR}\left( {j - 3} \right)}}}\end{matrix}}{b0}} & (39)\end{matrix}$

[0173] The manipulated variable calculator 31 shown in FIG. 7 representsa block diagram of the calculations according to the equation (39).

[0174] The feedback manipulated variable KSTR determined according tothe equation (39) becomes the target air-fuel ratio KCMD insofar as theoutput KACT of the LAF sensor 5 agrees with the target air-fuel ratioKCMD. Therefore, the feedback manipulated variable KSTR is divided bythe target air-fuel ratio KCMD by the divider 19 for thereby determiningthe feedback manipulated variable kstr that can be used as the feedbackcorrection coefficient KFB.

[0175] As is apparent from the foregoing description, the adaptivecontroller 18 thus constructed is a recursive-type controller takinginto account dynamic behavioral changes of the internal combustionengine 1 which is an object to be controlled. Stated otherwise, theadaptive controller 18 is a controller described in a recursive form tocompensate for dynamic behavioral changes of the internal combustionengine 1, and more particularly a controller having a recursive-typeadaptive parameter adjusting mechanism.

[0176] A recursive-type controller of this type may be constructed usingan optimum regulator. In such a case, however, it generally has noparameter adjusting mechanism. The adaptive controller 18 constructed asdescribed above is suitable for compensating for dynamic behavioralchanges of the internal combustion engine 1.

[0177] The details of the adaptive controller 18 have been describedabove.

[0178] The PID controller 17, which is provided together with theadaptive controller 18 in the general feedback controller 15, calculatesa proportional term (P term), an integral term (I term), and aderivative term (D term) from the difference between the output KACT ofthe LAF sensor 5 and the target air-fuel ratio KCMD, and calculates thetotal of those terms as the feedback manipulated variable KLAF, as isthe case with the general PID control process. In the presentembodiment, the feedback manipulated variable KLAF is set to “1” whenthe output KACT of the LAF sensor 5 agrees with the target air-fuelratio KCMD by setting an initial value of the integral term (I term) to“1”, so that the feedback manipulated variable KLAF can be used as thefeedback correction coefficient KFB for directly correcting the fuelinjection quantity. The gains of the proportional term, the integralterm, and the derivative term are determined from the rotational speedand intake pressure of the internal combustion engine 1 using apredetermined map.

[0179] The switcher 20 of the general feedback controller 15 outputs thefeedback manipulated variable KLAF determined by the PID controller 17as the feedback correction coefficient KFB for correcting the fuelinjection quantity if the combustion in the internal combustion engine 1tends to be unstable as when the temperature of the coolant of theinternal combustion engine 1 is low, the internal combustion engine 1rotates at high speeds, or the intake pressure is low, or if the outputKACT of the LAF sensor 5 is not reliable due to a response delay of theLAF sensor 5 as when the target air-fuel ratio KCMD changes largely orimmediately after the air-fuel ratio feedback control process hasstarted, or if the internal combustion engine 1 operates highly stablyas when it is idling and hence no high-gain control process by theadaptive controller 18 is required. Otherwise, the switcher 20 outputsthe feedback manipulated variable kstr which is produced by dividing thefeedback manipulated variable KSTR determined by the adaptive controller18 by the target air-fuel ration KCMD, as the feedback correctioncoefficient KFB for correcting the fuel injection quantity. This isbecause the adaptive controller 18 effects a high-gain control processand functions to converge the output KACT of the LAF sensor 5 quickly tothe target air-fuel ratio KCMD, and if the feedback manipulated variableKSTR determined by the adaptive controller 18 is used when thecombustion in the internal combustion engine 1 is unstable or the outputKACT of the LAF sensor 5 is not reliable, then the air-fuel ratiocontrol process tends to be unstable.

[0180] Such operation of the switcher 20 is disclosed in detail inJapanese laid-open patent publication No. 8-105345 by the applicant ofthe present application, and will not be described in detail below.

[0181] Operation of the apparatus according to the present embodimentwill be described below.

[0182] First, a process carried out by the engine-side control unit 7 bwill be described below with reference to FIG. 8. The engine-sidecontrol unit 7 b calculates an output fuel injection quantity #nTout foreach of the cylinders in synchronism with a crankshaft angle period(TDC) of the internal combustion engine 1 as follows:

[0183] The engine-side control unit 7 b reads outputs from varioussensors including the LAF sensor 5 and the O₂ sensor 6 in STEPa. At thistime, the output KACT of the LAF sensor 5 and the output VO2/OUT of theO₂ sensor 6, including data obtained in the past, are stored in atime-series fashion in a memory (not shown).

[0184] Then, the basic fuel injection quantity calculator 8 corrects afuel injection quantity corresponding to the rotational speed NE andintake pressure PB of the internal combustion engine 1 depending on theeffective opening area of the throttle valve, thereby calculating abasic fuel injection quantity Tim in STEPb. The first correctioncoefficient calculator 9 calculates a first correction coefficientKTOTAL depending on the coolant temperature and the amount by which thecanister is purged in STEPc.

[0185] The engine-side control unit 7 b decides whether the operationmode of the internal combustion engine 1 is an operation mode(hereinafter referred to as “normal operation mode”) in which the fuelinjection quantity is adjusted using the target air-fuel ratio KCMDgenerated by the target air-fuel ratio generation processor 13, and setsa value of a flag f/prism/on in STEPd. When the value of the flagf/prism/on is “1”, it means that the operation mode of the internalcombustion engine 1 is the normal operation mode, and when the value ofthe flag f/prism/on is “0”, it means that the operation mode of theinternal combustion engine 1 is not the normal operation mode.

[0186] The deciding subroutine of STEPd is shown in detail in FIG. 9. Asshown in FIG. 9, the engine-side control unit 7 b decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEPd-1, STEPd-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the target air-fuel ratio generation processor 13are not accurate enough, the operation mode of the internal combustionengine 1 is not the normal operation mode, and the value of the flagf/prism/on is set to “0” in STEPd-10.

[0187] Then, the engine-side control unit 7 b decides whether theinternal combustion engine 1 is operating with a lean air-fuel mixtureor not in STEPd-3. The engine-side control unit 7 b decides whether theignition timing of the internal combustion engine 1 is retarded forearly activation of the catalytic converter 3 immediately after thestart of the internal combustion engine 1 or not in STEPd-4. Theengine-side control unit 7 b decides whether the throttle valve of theinternal combustion engine 1 is substantially fully open or not inSTEPd-5. The engine-side control unit 7 b decides whether the supply offuel to the internal combustion engine 1 is being stopped or not inSTEPd-6. If either one of the conditions of these steps is satisfied,then since it is not preferable or not possible to control the supply offuel to the internal combustion engine 1 using the target air-fuel ratioKCMD generated by the target air-fuel ratio generation processor 13, theoperation mode of the internal combustion engine 1 is not the normaloperation mode, and the value of the flag f/prism/on is set to “0” inSTEPd-10.

[0188] The engine-side control unit 7 b then decides whether therotational speed NE and the intake pressure PB of the internalcombustion engine 1 fall within respective given ranges (normal ranges)or not respectively in STEPd-7, STEPd-8. If either one of the rotationalspeed NE and the intake pressure PB does not fall within its givenrange, then since it is not preferable to control the supply of fuel tothe internal combustion engine 1 using the target air-fuel ratio KCMDgenerated by the target air-fuel ratio generation processor 13, theoperation mode of the internal combustion engine 1 is not the normaloperation mode, and the value of the flag f/prism/on is set to “0” inSTEPd-10.

[0189] If the conditions of STEPd-1, STEPd-2, STEPd-7, STEPd-8 aresatisfied, and the conditions of STEPd-3, STEPd-4, STEPd-5, STEPd-6 arenot satisfied (at this time, the internal combustion engine 1 is in thenormal operation mode), then the operation mode of the internalcombustion engine 1 is judged as the normal operation mode, and thevalue of the flag f/prism/on is set to “1” in STEPd-9.

[0190] In FIG. 8, after the value of the flag f/prism/on has been set,the engine-side control unit 7 b determines the value of the flagf/prism/on in STEPe. If f/prism/on=1, then the engine-side control unit7 b reads the target air-fuel ratio KCMD generated by the exhaust-sidemain processor 13 in STEPf. If f/prism/on=0, then the engine-sidecontrol unit 7 b sets the target air-fuel ratio KCMD to a predeterminedvalue in STEPg. The predetermined value to be established as the targetair-fuel ratio KCMD is determined from the rotational speed NE andintake pressure PB of the internal combustion engine 1 using apredetermined map, for example.

[0191] In the local feedback controller 16, the PID controller 22calculates respective feedback correction coefficients #nKLAF in orderto eliminate variations between the cylinders, based on actual air-fuelratios #nA/F of the respective cylinders which have been estimated fromthe output KACT of the LAF sensor 5 by the observer 21, in STEPh. Then,the general feedback controller 15 calculates a feedback correctioncoefficient KFB in STEPi.

[0192] Depending on the operating state of the internal combustionengine 1, the switcher 20 selects either the feedback manipulatedvariable KLAF determined by the PID controller 17 or the feedbackmanipulated variable kstr which has been produced by dividing thefeedback manipulated variable KSTR determined by the adaptive controller18 by the target air-fuel ratio KCMD (normally, the switcher 20 selectsthe feedback manipulated variable kstr from the adaptive controller 18).The switcher 20 then outputs the selected feedback manipulated variableKLAF or kstr as a feedback correction coefficient KFB for correcting thefuel injection quantity.

[0193] When switching the feedback correction coefficient KFB from thefeedback manipulated variable KLAF from the PID controller 17 to thefeedback manipulated variable kstr from the adaptive controller 18, theadaptive controller 18 determines a feedback manipulated variable KSTRin a manner to hold the correction coefficient KFB to the precedingcorrection coefficient KFB (=KLAF) as long as in the cycle time for theswitching. When switching the feedback correction coefficient KFB fromthe feedback manipulated variable kstr from the adaptive controller 18to the feedback manipulated variable KLAF from the PID controller 17,the PID controller 17 calculates a present correction coefficient KLAFin a manner to regard the feedback manipulated variable KLAF determinedby itself in the preceding cycle time as the preceding correctioncoefficient KFB (=kstr).

[0194] After the feedback correction coefficient KFB has beencalculated, the second correction coefficient calculator 10 calculatesin STEPj a second correction coefficient KCMDM depending on the targetair-fuel ratio KCMD determined in STEPf or STEPg.

[0195] Then, the engine-side control unit 7 b multiplies the basic fuelinjection quantity Tim determined as described above, by the firstcorrection coefficient KTOTAL, the second correction coefficient KCMDM,the feedback correction coefficient KFB, and the feedback correctioncoefficients #nKLAF of the respective cylinders, determining output fuelinjection quantities #nTout of the respective cylinders in STEPk. Theoutput fuel injection quantities #nTout are then corrected foraccumulated fuel particles on intake pipe walls of the internalcombustion engine 1 by the fuel accumulation corrector 23 in STEPm. Thecorrected output fuel injection quantities #nTout are output to thenon-illustrated fuel injectors of the internal combustion engine 1 inSTEPn. In the internal combustion engine 1, the fuel injectors injectfuel into the respective cylinders according to the respective outputfuel injection quantities #nTout.

[0196] The above calculation of the output fuel injection quantities#nTout and the fuel injection of the internal combustion engine 1 arecarried out in successive cycle times synchronous with the crankshaftangle period of the internal combustion engine 1 for controlling theair-fuel ratio of the internal combustion engine 1 in order to convergethe output KACT of the LAF sensor 5 (the detected value of theupstream-of-catalyst air-fuel ratio) to the target air-fuel ratio KCMD.While the feedback manipulated variable kstr from the adaptivecontroller 18 is being used as the feedback correction coefficient KFB,the output KACT of the LAF sensor 5 is quickly converged to the targetair-fuel ratio KCMD with high stability against behavioral changes suchas changes in the operating state of the internal combustion engine 1 orcharacteristic changes thereof. A response delay of the internalcombustion engine 1 is also appropriately compensated for.

[0197] Concurrent with the above fuel control for the internalcombustion engine 1, the exhaust-side control unit 7 a executes aflowchart of FIG. 13 in control cycles of a constant period.

[0198] As shown in FIG. 13, the exhaust-side control unit 7 a decideswhether the processing of the target air-fuel ratio generation processor13 (specifically, the processing of the identifier 25, the estimator 26,and the sliding mode controller 27) is to be executed or not, and sets avalue of a flag f/prism/cal indicative of whether the processing is tobe executed or not in STEP1. When the value of the flag f/prism/cal is“0”, it means that the processing of the target air-fuel ratiogeneration processor 13 is not to be executed, and when the value of theflag f/prism/cal is “1”, it means that the processing of the targetair-fuel ratio generation processor 13 is to be executed.

[0199] The deciding subroutine in STEP1 is shown in detail in FIG. 11.As shown in FIG. 11, the exhaust-side control unit 7 a decides whetherthe O₂ sensor 6 and the LAF sensor 5 are activated or not respectivelyin STEP1-1, STEP1-2. If neither one of the O₂ sensor 6 and the LAFsensor 5 is activated, since detected data from the O₂ sensor 6 and theLAF sensor 5 for use by the target air-fuel ratio generation processor13 are not accurate enough, the value of the flag f/prism/cal is set to“0” in STEP1-6. Then, in order to initialize the identifier 25 asdescribed later on, the value of a flag f/id/reset indicative of whetherthe identifier 25 is to be initialized or not is set to “1” in STEP1-7.When the value of the flag f/id/reset is “1”, it means that theidentifier 25 is to be initialized, and when the value of the flagf/id/reset is “0”, it means that the identifier 25 is not to beinitialized.

[0200] The exhaust-side control unit 7 a decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEP1-3. The exhaust-side control unit 7 a decides whether the ignitiontiming of the internal combustion engine 1 is retarded for earlyactivation of the catalytic converter 3 immediately after the start ofthe internal combustion engine 1 or not in STEP1-4. If the conditions ofthese steps are satisfied, then since the target air-fuel ratio KCMDcalculated to adjust the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET is not used for the fuel control for the internalcombustion engine 1, the value of the flag f/prism/cal is set to “0” inSTEP1-6, and the value of the flag f/id/reset is set to “1” in order toinitialize the identifier 25 in STEP1-7.

[0201] In FIG. 10, after the above deciding subroutine, the exhaust-sidecontrol unit 7 a decides whether a process of identifying (updating) thegain coefficients a1, a1, b1 with the identifier 25 is to be executed ornot, and sets a value of a flag f/id/cal indicative of whether theprocess of identifying (updating) the gain coefficients a1, a1, b1 is tobe executed or not in STEP2. When the value of the flag f/id/cal is “0”,it means that the process of identifying (updating) the gaincoefficients a1, a1, b1 is not to be executed, and when the value of theflag f/id/cal is “1”, it means that the process of identifying(updating) the gain coefficients a1, a1, b1 is to be executed.

[0202] In the deciding process of STEP2, the exhaust- side control unit7 a decides whether the throttle valve of the internal combustion engine1 is substantially fully open or not, and also decides whether thesupply of fuel to the internal combustion engine 1 is being stopped ornot. If either one of these conditions is satisfied, then since it isdifficult to identify the gain coefficients a1, a1, b1 appropriately,the value of the flag f/id/cal is set to “0”. If neither one of theseconditions is satisfied, then the value of the flag f/id/cal is set to“1” to identify (update) the gain coefficients a1, a1, b1 with theidentifier 25.

[0203] The flow rate data generating means 28 calculates an estimatedexhaust gas volume ABSV according to the equation (3) from the latestdetected values (acquired by the engine-side control unit 7 b in STEPain FIG. 8) of the present rotational speed NE and intake pressure PB ofthe internal combustion engine 1 in STEP3. Thereafter, the dead timesetting means 29 determines the values of respective set dead times d1,d2 of the exhaust system E and the air-fuel ratio manipulating systemfrom the calculated value of the estimated exhaust gas volume ABSVaccording to the data table shown in FIG. 4 in STEP4. The values of theset dead times d1, d2 determined in STEP4 are integral values which areproduced by rounding off the fractions of the values determined from thedata table shown in FIG. 4, as described above.

[0204] Then, the exhaust-side control unit 7 a calculates the latestdifferential outputs kact(k) (=KACT−FLAF/BASE), VO2(k)(=VO2/OUT−VO2/TARGET) respectively with the subtractors 11, 12 in STEP5.Specifically, the subtractors 11, 12 select latest ones of thetime-series data read and stored in the non-illustrated memory in STEPashown in FIG. 8, and calculate the differential outputs kact(k), VO2(k).The differential outputs kact(k), VO2(k), as well as data given in thepast, are stored in a time-series manner in the non-illustrated memoryin the exhaust-side control unit 7 a.

[0205] Then, in STEP6, the exhaust-side control unit 7 a determines thevalue of the flag f/prism/cal set in STEP1. If the value of the flagf/prism/cal is “0”, i.e., if the processing of the target air-fuel ratiogeneration processor 13 is not to be executed, then the exhaust-sidecontrol unit 7 a forcibly sets the SLD manipulating input Usl (thetarget differential air-fuel ratio kcmd) to be determined by the slidingmode controller 27, to a predetermined value in STEP14. Thepredetermined value may be a fixed value (e.g., “0”) or the value of theSLD manipulating input Usl determined in a preceding control cycle.

[0206] After the SLD manipulating input Usl is set to the predeterminedvalue, the exhaust-side control unit 7 a adds the reference valueFLAF/BASE to the SLD manipulating input Usl for thereby determining atarget air-fuel ratio KCMD in the present control cycle in STEP 15.Then, the processing in the present control cycle is finished.

[0207] If the value of the flag f/prism/cal is “1” in STEP6, i.e., ifthe processing of the target air-fuel ratio generation processor 13 isto be executed, then the exhaust-side control unit 7 a effects theprocessing of the identifier 25 in STEP7.

[0208] The processing of the identifier 25 is carried out according to aflowchart shown in FIG. 12. The identifier 25 determines the value ofthe flag f/id/cal set in STEP2 in STEP7-1. If the value of the flagf/id/cal is “0”, then since the process of identifying the gaincoefficients a1, a1, b1 with the identifier 25 is not carried out,control immediately goes back to the main routine shown in FIG. 10.

[0209] If the value of the flag f/id/cal is “1”, then the identifier 25determines the value of the flag f/id/reset set in STEP1 with respect tothe initialization of the identifier 25 in STEP7-2. If the value of theflag f/id/reset is “1”, the identifier 25 is initialized in STEP7-3.When the identifier 25 is initialized, the identified gain coefficientsa1 hat, a2 hat, b1 hat are set to predetermined initial values (theidentified gain coefficient vector Θ according to the equation (5) isinitialized), and the elements of the matrix P (diagonal matrix)according to the equation (11) are set to predetermined initial values.The value of the flag f/id/reset is reset to “0”.

[0210] Then, the identifier 25 determines the value of the weightedparameter λ₁ in the algorithm of the method of weighted least squares ofthe identifier 25, i.e., the value of the weighted parameter λ₁ used inthe equation (11), from the present value of the estimated exhaust gasvolume ABSV determined by the flow rate data generating means 28 inSTEP3 according to the data table shown in FIG. 5 in STEP7-4.

[0211] Then, the identifier 25 calculates the identified differentialoutput VO2(k) hat using the values of the present identified gaincoefficients a1(k−1) hat, a2(k−1) hat, b1(k−1) hat and the past dataVO2(k−1), VO2(k−2), kact(k−d1−1) of the differential outputs VO2, kactcalculated in each control cycle in STEP5, according to the equation (4)in STEP7-5. Specifically, the differential output kact(k−d1−1) used inthe above calculation is a differential output kact at a past timedetermined by the set dead time d1 of the exhaust system E that is setby the dead time setting means 29 in STEP4, and also a differentialoutput kact obtained in a control cycle that is (d1+1) control cyclesprior to the present control cycle.

[0212] The identifier 25 then calculates the vector Kθ (k) to be used indetermining the new identified gain coefficients a1 hat, a2 hat, b1 hataccording to the equation (10) in STEP7-6. Thereafter, the identifier 25calculates the identified error id/e(k) (the difference between theidentified differential output VO2 hat and the actual differentialoutput VO2, see the equation (8)), in STEP7-7.

[0213] The identified error id/e(k) may basically be calculatedaccording to the equation (8). In the present embodiment, however, avalue (=VO2(k)−VO2(k) hat) calculated according to the equation (8) fromthe differential output VO2 calculated in each control cycle in STEP3,and the identified differential output VO2 hat calculated in eachcontrol cycle in STEP7-5 is filtered with low-pass characteristics tocalculate the identified error id/e(k).

[0214] This is because since the behavior of the exhaust system Eincluding the catalytic converter 3 generally has low-passcharacteristics, it is preferable to attach importance to thelow-frequency behavior of the exhaust system E in appropriatelyidentifying the gain coefficients a1, a2, b1 of the exhaust systemmodel.

[0215] Both the differential output VO2 and the identified differentialoutput VO2 hat may be filtered with the same low-pass characteristics.For example, after the differential output VO2 and the identifieddifferential output VO2 hat have separately been filtered, the equation(7) may be calculated to determine the identified error id/e(k). Theabove filtering is carried out by a moving average process which is adigital filtering process.

[0216] Thereafter, the identifier 25 calculates a new identified gaincoefficient vector Θ (k), i.e., new identified gain coefficients a1(k)hat, a2(k) hat, b1(k) hat, according to the equation (9) using theidentified error id/e(k) determined in STEP7-7 and KO calculated inSETP7-6 in STEP7-8.

[0217] After having calculated the new identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat, the identifier 25 limits the values ofthe gain coefficients a1 hat, a2 hat, b1 hat within a predeterminedrange as described below in STEP7-9. Then, the identifier 25 updates thematrix P(k) according to the equation (11) for the processing of a nextcontrol cycle in STEP7-10, after which control returns to the mainroutine shown in FIG. 10.

[0218] The process of limiting the identified gain coefficients a1 hat,a2 hat, b1 hat in STEP7-9 comprises a process of eliminating thesituation where the target air-fuel ratio KCMD determined by the slidingmode controller 27 varies in a high-frequency oscillating manner. Theinventors of the present invention have found that if the values of theidentified gain coefficients a1 hat, a2 hat, b1 hat are not particularlylimited, while the output signal VO2/OUT of the O₂ sensor 6 is beingstably controlled at the target value VO2/TARGET, there are developed asituation in which the target air-fuel ratio KCMD determined by thesliding mode controller 27 changes smoothly with time, and a situationin which the target air-fuel ratio KCMD oscillates with time at a highfrequency. Whether the target air-fuel ratio KCMD changes smoothly oroscillates at a high frequency depends on the combinations of the valuesof the identified gain coefficients a1 hat, a2 hat relative to theresponse delay element of the exhaust system model (more specifically,the primary autoregressive term and the secondary autoregressive term onthe right side of the equation (1)) and the value of the identified gaincoefficient b1 hat relative to the dead time element of the exhaustsystem model.

[0219] The limiting process in STEP7-9 is roughly classified into aprocess of limiting the combination of the values of the identified gaincoefficients a1 hat, a2 hat within a given range, and a process oflimiting the value of the identified gain coefficient b1 hat within agiven range.

[0220] The range within which the combination of the values of theidentified gain coefficients a1 hat, a2 hat are limited and the rangewithin which the value of the identified gain coefficient b1 hat islimited are established as follows:

[0221] With respect to the range within which the combination of thevalues of the identified gain coefficients a1 hat, a2 hat are limited, astudy made by the inventors indicates that whether the target air-fuelratio KCMD changes smoothly or oscillates at a high frequency is closelyrelated to combinations of the coefficient values a1, a2 used for theestimator 26 to determine the estimated differential output VO2(k+d) bar(these coefficient values α1, α2 are the first-row, first-column elementand the first-row, second-column element of the matrix A^(d) which is apower of the matrix A defined by the equation (13)).

[0222] Specifically, as shown in FIG. 13, when a coordinate plane whosecoordinate components are represented by the coefficient values α1, α2is established, if a point on the coordinate plane which is determinedby a combination of the coefficient values α1, α2 lies in a hatchedrange, which is surrounded by a triangle Q₁Q₂Q₃ (including theboundaries) and will hereinafter be referred to as an estimatingcoefficient stable range, then the target air-fuel ratio KCMD tends tobe smooth. Conversely, if a point determined by a combination of thecoefficient values α1, α2 lies outside of the estimating coefficientstable range, then the target air-fuel ratio KCMD is liable to oscillatewith time at a high frequency or the controllability of the outputVO2/OUT of the O₂ sensor 6 at the target value VO2/TARGET is liable tobecome poor.

[0223] Therefore, the combinations of the values of the gaincoefficients a1, a2 should be limited such that the point on thecoordinate plane shown in FIG. 13 which corresponds to the combinationof the coefficient values α1, α2 determined by the values of theidentified gain coefficients a1 hat, a2 hat will lie within theestimating coefficient stable range.

[0224] In FIG. 13, a triangular range Q₁Q₄Q₃ on the coordinate planewhich contains the estimating coefficient stable range is a range thatdetermines combinations of the coefficient values α1, α2 which makestheoretically stable a system defined according to the followingequation (40), i.e., a system defined by an equation similar to theequation (13) except that VO2(k), VO2(k−1) on the right side of theequation (13) are replaced respectively with VO2(k) bar, VO2(k−1) bar(VO2(k) bar, VO2(k−1) bar mean respectively an estimated differentialoutput determined in each control cycle by the estimator 26 and anestimated differential output determined in a preceding cycle by theestimator 26). $\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{{\alpha 1} \cdot {\overset{\_}{VO2}(k)}} + {{\alpha 2} \cdot {\overset{\_}{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{d}{\beta_{j} \cdot {{kcmd}\left( {k - j} \right)}}}}} & (40)\end{matrix}$

[0225] The condition for the system defined according to the equation(40) to be stable is that a pole of the system (which is given by thefollowing equation (41)) exists in a unit circle on a complex plane:$\begin{matrix}{\text{Pole~~of~~the~~system~~according~~to~~the~~equation~~(40)} = \frac{{\alpha 1} \pm \sqrt{{\alpha 1}^{2} + {4 \cdot {\alpha 2}}}}{2}} & (41)\end{matrix}$

[0226] The triangular range Q₁Q₄Q₃ shown in FIG. 13 is a range fordetermining the combinations of the coefficient values α1, α2 whichsatisfy the above condition. Therefore, the estimating coefficientstable range is a range indicative of those combinations where α1≧0 ofthe combinations of the coefficient values α1, α2 which make stable thesystem defined by the equation (40).

[0227] Since the coefficient values α1, α2 are determined by acombination of the values of the gain coefficients a1, a2 when the totalset dead time d is determined to be of a certain value, a combination ofthe values of the gain coefficients a1, a2 is determined from acombination of the coefficient values α1, α2 using the value of thetotal set dead time d. Therefore, the estimating coefficient stablerange shown in FIG. 13 which determines preferable combinations of thecoefficient values α1, α2 can be converted into a range on a coordinateplane shown in FIG. 14 whose coordinate components are represented bythe gain coefficients a1, a2.

[0228] If the above conversion is carried out with the total set deadtime d being determined to be of a certain value, then the estimatingcoefficient stable range is converted into a range enclosed by theimaginary lines in FIG. 14, which is a substantially triangular rangehaving an undulating lower side and will hereinafter be referred to asan identifying coefficient stable range, on the coordinate plane shownin FIG. 14. Stated otherwise, when a point on the coordinate plane shownin FIG. 14 which is determined by a combination of the values of thegain coefficients a1, a2 resides in the identifying coefficient stablerange enclosed by the imaginary lines in FIG. 14, a point on thecoordinate plane shown in FIG. 13 which corresponds to the combinationof the coefficient values α1, α2 determined by those values of the gaincoefficients a1, a2 resides in the estimating coefficient stable range.The identifying coefficient stable range changes with the value of thetotal set dead time d, as described later on. It is assumed for a whilein the description below that the total set dead time d is fixed to acertain value (represented by dx).

[0229] Consequently, the combinations of the values of the identifiedgain coefficients a1 hat, a2 hat determined by the identifier 25 shouldpreferably be limited within such a range that a point on the coordinateplane shown in FIG. 14 which is determined by those values of theidentified gain coefficients a1 hat, a2 hat reside in the identifyingcoefficient stable range.

[0230] However, since a boundary (lower side) of the identifyingcoefficient stable range indicated by the imaginary lines in FIG. 14 isof a complex undulating shape, a practical process for limiting thepoint on the coordinate plane shown in FIG. 14 which is determined bythe values of the identified gain coefficients a1 hat, a2 hat within theidentifying coefficient stable range is liable to be complex.

[0231] For this reason, according to the present embodiment, theidentifying coefficient stable range (the identifying coefficient stablerange corresponding to the total set dead time dx) is substantiallyapproximated by a quadrangular range Q₅Q₆Q₇Q₈ enclosed by the solidlines in FIG. 14, which has straight boundaries and will hereinafter bereferred to as an identifying coefficient limiting range. As shown inFIG. 14, the identifying coefficient limiting range (the identifyingcoefficient limiting range corresponding to the total set dead time dx)is a range enclosed by a polygonal line (including line segments Q₅Q₆and Q₅Q₈) expressed by a functional expression |a1|+a2=1, a straightline (including a line segment Q₆Q₇) expressed by a constant-valuedfunctional expression a1=A1L, and a straight line (including a linesegment Q₇Q₈) expressed by a constant-valued functional expressiona2=A2L. In the present embodiment, the identifying coefficient limitingrange is used as the range within which the combinations of the valuesof the identified gain coefficients a1 hat, a2 hat are limited. Althoughpart of the lower side of the identifying coefficient limiting rangedeviates from the identifying coefficient stable range, it hasexperimentally been confirmed that the point determined by theidentified gain coefficients a1 hat, a2 hat determined by the identifier25 does not actually fall in the deviating range. Therefore, thedeviating range will not pose any practical problem.

[0232] The identifying coefficient stable range which serves as a basisfor the identifying coefficient limiting range changes with the value ofthe total set dead time d, as is apparent from the definition of thecoefficient values α1, α2 according to the equation (13). In the presentembodiment, the values of the set dead time d1 of the exhaust system Eand the set dead time d2 of the air-fuel ratio manipulating system, andhence the value of the total set dead time d, are sequentially variablyset depending on the estimated exhaust gas volume ABSV.

[0233] The inventors have found that the identifying coefficient stablerange, chiefly the shape of only its lower portion (generally anundulating portion from Q7 to Q8 in FIG. 14), varies depending on thevalue of the total set dead time d, and as the value of the total setdead time d is greater, the lower portion of the identifying coefficientstable range tends to expand more downwardly (in the negative directionalong the a2 axis). The shape of the upper portion (generally a portionenclosed by a triangle Q5Q6Q8 in FIG. 14) of the identifying coefficientstable range is almost not affected by the value of the total set deadtime d.

[0234] In the present embodiment, the lower limit value A2L of the gaincoefficient al in the identifying coefficient limiting range forlimiting the combinations of the values of the identified gaincoefficients a1 hat, a2 hat is variably set depending on the estimatedexhaust gas volume ABSV which determines the dead times d1, d2 of theexhaust system E and the air-fuel ratio manipulating system. The lowerlimit value A2L of the gain coefficient al is determined from the value(latest value) of the estimated exhaust gas volume ABSV based on apredetermined data table represented by the solid-line curve e in FIG.15, for example. The data table is determined such that as the value ofthe estimated exhaust gas volume ABSV is larger (as the total set deadtime d is shorter), the lower limit value A2L (<0) is smaller (theabsolute value is greater). Thus, the identifying coefficient limitingrange is established such that as the estimated exhaust gas volume ABSVis larger (as the total set dead time d is shorter), the identifyingcoefficient limiting range is expanded more downwardly. For example, ifthe value of the total set dead time d is longer than the value dxcorresponding to the identifying coefficient limiting range indicated bythe solid line in FIG. 14, then the lower portion of the identifyingcoefficient limiting range is expanded below the identifying coefficientlimiting range where d=dx.

[0235] The above identifying coefficient limiting range is given forillustrative purpose only, and may be equal to or may substantiallyapproximate the identifying coefficient stable range corresponding toeach value of the total set dead time d, or may be of any shape insofaras most or all of the identifying coefficient limiting range belongs tothe identifying coefficient stable range. Thus, the identifyingcoefficient limiting range may be established in various configurationsin view of the ease with which to limit the values of the identifiedgain coefficients a1 hat, a2 hat and the practical controllability. Forexample, while the boundary of an upper portion of the identifyingcoefficient limiting range is defined by the functional expression|a1|+a2=1 in the illustrated embodiment, combinations of the values ofthe gain coefficients a1, a2 which satisfy this functional expressionare combinations of theoretical stable limits where a pole of the systemdefined by the equation (40) exists on a unit circle on a complex plane.Therefore, the boundary of the upper portion of the identifyingcoefficient limiting range may be determined by a functional expression|a1|+a2=r (r is a value slightly smaller than “1” corresponding to thestable limits, e.g., 0.99) for higher control stability.

[0236] The range within which the value of the identified gaincoefficient b1 hat is limited is established as follows:

[0237] The inventors have found that the situation in which thetime-depending change of the target air-fuel ratio KCMD is oscillatoryat a high frequency tends to happen also when the value of theidentified gain coefficient b1 hat is excessively large or small.Furthermore, the value of the identified gain coefficient b1 hat whichis suitable to cause the target air-fuel ratio KCMD to change smoothlywith time is affected by the total set dead time d, and tends to begreater as the total set dead time d is shorter. According to thepresent embodiment, an upper limit value B1H and a lower limit value B1L(B1H>B1L>0) for determining the range of the gain coefficient b1 aresequentially established depending on the value (latest value) of theestimated exhaust gas volume ABSV which determines the value of thetotal set dead time d, and the value of the identified gain coefficientb1 hat is limited in a range that is determined by the upper limit valueB1H and the lower limit value B1L. In the present embodiment, the upperlimit value B1H and the lower limit value B1L which determine the rangeof the value of the gain coefficient b1 are determined based on datatables that are determined in advance through experimentation orsimulation as indicated by the solid-line curves f, g in FIG. 15. Thedata tables are basically established that as the estimated exhaust gasvolume ABSV is greater (as the total set dead time d is shorter), theupper limit value B1H and the lower limit value B1L are greater.

[0238] A process of limiting combinations of the values of theidentified gain coefficients a1 hat, a2 hat and the range of the valueof the identified gain coefficient b1 is carried out as follows:

[0239] Referring to a flowchart shown in FIG. 16, the identifier 25 setsthe lower limit value A2L of the gain coefficient a2 in the identifyingcoefficient limiting range and the upper limit value B1H and the lowerlimit value B1L of the gain coefficient b1 based on the data tablesshown in FIG. 15 from the latest value of the estimated exhaust gasvolume ABSV determined by the flow rate data generating means 28 inSTEP3 shown in FIG. 10, in STEP7-9-1.

[0240] The identifier 25 first limits combinations of the identifiedgain coefficients a1(k) hat, a2(k) hat, of the identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat that have been determinedin STEP7-8 shown in FIG. 12, within the identifying coefficient limitingrange in STEP7-9-2 through STEP7-9-9.

[0241] Specifically, the identifier 25 decides whether or not the valueof the identified gain coefficient a2(k) hat determined in STEP7-8 isequal to or greater than the lower limit value A2L (see FIG. 14) set inSTEP7-9-1, in STEP7-9-2.

[0242] If A2(k) hat<A2L, then since a point on the coordinate planeshown in FIG. 14 (expressed by (a1(k) hat, a2(k) hat), determined by thecombination of the values of the identified gain coefficients a1(k) hat,a2(k) hat does not reside in the identifying coefficient limiting range,the value of a2(k) hat is forcibly changed to the lower limit value A2Lin STEP7-9-3. Thus, the point (a1(k) hat, a2(k) hat) on the coordinateplane shown in FIG. 14 is limited to a point in a region on and above astraight line (the straight line including the line segment Q₇Q₈)expressed by at least a2=A2L.

[0243] Then, the identifier 25 decides whether or not the value of theidentified gain coefficient a1(k) hat determined in STEP7-8 is equal toor greater than a lower limit value A1L (see FIG. 14) for the gaincoefficient a1 in the identifying coefficient limiting range inSTEP7-9-4, and then decides whether or not the value of the identifiedgain coefficient a1(k) hat is equal to or smaller than an upper limitvalue A1H (see FIG. 14) for the gain coefficient al in the identifyingcoefficient limiting range in STEP7-9-6. In the present embodiment, thelower limit value A1L for the gain coefficient al is a predeterminedfixed value. The upper limit value A1H for the gain coefficient a1 isrepresented by A1H=1−A2L because it is an al coordinate of the point Q₈where the polygonal line |a1|+a2=1 (a1>0) and the straight line a2=A2Lintersect with each other, as shown in FIG. 14.

[0244] If a1(k) hat<A1L or a1(k) hat>A1H, then since the point (a1(k)hat, a2(k) hat) on the coordinate plane shown in FIG. 14 does not residein the identifying coefficient limiting range, the value of a1(k) hat isforcibly changed to the lower limit value A1L or the upper limit valueA1H in STEP7-9-5 and STEP7-9-7.

[0245] Thus, the point (a1(k) hat, a2(k) hat) on the co-ordinate planeshown in FIG. 14 is limited to a region on and between a straight line(the straight line including the line segment Q₆Q₇) expressed by a1=A1L,and a straight line (the straight line passing through the point Q₈ andperpendicular to the a1 axis) expressed by a1=A1H.

[0246] The processing in STEP7-9-4 through STEP7-9-7 may be carried outbefore the processing in STEP7-9-2 and STEP7-9-3.

[0247] Then, the identifier 25 decides whether the present values ofa1(k) hat, a2(k) hat after STEP7-9-2 through STEP7-9-7 satisfy aninequality |a1|+a2≦1 or not, i.e., whether the point (a1(k) hat, a2(k)hat) is positioned on or below or on or above the polygonal line(including line segments Q₅Q₆ and Q₅Q₈) expressed by the functionalexpression |a1|+a2=1 in STEP7-9-8.

[0248] If |a1|+a2≦1, then the point (a1(k) hat, a2(k) hat) determined bythe values of a1(k) hat, a2(k) hat after the processing in STEP7-9-2through STEP7-9-7 exists in the identifying coefficient limiting range(including its boundaries).

[0249] If |a1|+a2>1, then since the point (a1(k) hat, a2(k) hat)deviates upwardly from the identifying coefficient limiting range, thevalue of the a2(k) hat is forcibly changed to a value (1−|a1(k) hat|)depending on the value of a1(k) hat in STEP7-9-9. Stated otherwise,while the value of a1(k) hat is being kept unchanged, the point (a1(k)hat, a2(k) hat) is moved onto a polygonal line expressed by thefunctional expression |a1|+a2=1 (onto the line segment Q₅Q₆ or the linesegment Q₅Q₈ which is a boundary of the identifying coefficient limitingrange).

[0250] Through the above processing in STEP7-9-2 through 7-9-9, thevalues of the identified gain coefficients a1(k) hat, a2(k) hat arelimited such that the point (a1(k) hat, a2(k) hat) determined therebyresides in the identifying coefficient limiting range. If the point(a1(k) hat, a2(k) hat) corresponding to the values of the identifiedgain coefficients a1(k) hat, a2(k) hat that have been determined inSTEP7-8 exists in the identifying coefficient limiting range, then thosevalues of the identified gain coefficients a1(k) hat, a2(k) hat aremaintained.

[0251] The value of the identified gain coefficient a1(k) hat relativeto the primary autoregressive term of the discrete-system model is notforcibly changed insofar as the value resides between the lower limitvalue A1L and the upper limit value A1H of the identifying coefficientlimiting range. If a1(k) hat<A1L or a1(k) hat>A1H, then since the valueof the identified gain coefficient a1(k) hat is forcibly changed to thelower limit value A1L which is a minimum value that the gain coefficiental can take in the identifying coefficient limiting range or the upperlimit value A1H which is a maximum value that the gain coefficient alcan take in the identifying coefficient limiting range, the change inthe value of the identified gain coefficient a1(k) hat is minimum.Stated otherwise, if the point (a1(k) hat, a2(k) hat) corresponding tothe values of the identified gain coefficients a1(k) hat, a2(k) hat thathave been determined in STEP7-8 deviates from the identifyingcoefficient limiting range, then the forced change in the value of theidentified gain coefficient a1(k) hat is held to a minimum.

[0252] After having limited the values of the identified gaincoefficients a1(k) hat, a2(k) hat, the identifier 25 performs a processof limiting the value of the identified gain coefficient b1(k) hat inSTEP7-9-10 through STEP7-9-13.

[0253] Specifically, the identifier 25 decides whether or not the valueof the identified gain coefficient b1(k) hat determined in STEP7-8 isequal to or greater than the lower limit value B1L for the gaincoefficient b1 set in STEP7-9-1 in STEP7-9-10. If B1L>b1(k) hat, thenthe value of b1(k) hat is forcibly changed to the lower limit value B1Lin STEP7-9-11.

[0254] The identifier 25 decides whether or not the value of theidentified gain coefficient b1(k) hat is equal to or smaller than theupper limit value B1H for the gain coefficient g1 set in STEP7-9-1 inSTEP7-9-12. If B1H<b1(k) hat, then the value of b1(k) hat is forciblychanged to the upper limit value B1H in STEP7-9-13.

[0255] Through the above processing in STEP7-9-10 through 7-9-13, thevalue of the identified gain coefficient b1(k) hat is limited to a valuein a range between the lower limit value B1L and the upper limit valueB1H.

[0256] After the identifier 25 has limited the combination of the valuesof the identified gain coefficients a1(k) hat, a2(k) hat and theidentified gain coefficient b1(k) hat, control returns to the flowchartshown in FIG. 12.

[0257] The preceding values a1(k−1) hat, a2(k−1) hat, b1(k−1) hat of theidentified gain coefficients used for determining the identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat in STEP7-8 shown in FIG. 12are the values of the identified gain coefficients limited by thelimiting process in STEP7-9 in the preceding control cycle.

[0258] The above process is the processing sequence of the identifier 25which is carried out in STEP7 shown in FIG. 10.

[0259] In FIG. 10, after the processing of the identifier 25 has beencarried out, the exhaust-side control unit 7 a determines the values ofthe gain coefficients a1, a2, b1 in STEP8. Specifically, if the value ofthe flag f/id/cal set in STEP2 is “1”, i.e., if the gain coefficientsa1, a2, b1 have been identified by the identifier 25, then the gaincoefficients a1, a2, b1 are set to the latest identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat determined by theidentifier 25 in STEP7 (limited in STEP7-9). If f/id/cal=“0”, i.e., ifthe gain coefficients a1, a2, b1 have not been identified by theidentifier 25, then the gain coefficients a1, a2, b1 are set topredetermined values, respectively.

[0260] Then, the exhaust-side control unit 7 a effects a processingoperation of the estimator 26 in STEP9.

[0261] The estimator 26 calculates the coefficients a1, α2, βj (j=1, 2,. . . , d) to be used in the equation (14) or (15), using the gaincoefficients a1, a2, b1 determined in STEP8 (these values are basicallythe latest values of the identified gain coefficients a1 hat, a2 hat, b1hat) and the set dead time d1 of the exhaust system E and the set deadtime d2 of the air-fuel ratio manipulating system, which have been setin STEP4, according to the definition with respect to the equation (13).

[0262] Then, the estimator 26 calculates the estimated differentialoutput VO2(k+d) bar (estimated value of the differential output VO2after the total set dead time d from the time of the present controlcycle) according to the equation (14), using the time-series dataVO2(k), VO2(k−1) of the present and past values of the differentialoutput VO2 of the O₂ sensor calculated in each control cycle in STEP5,the time-series data kact(k−j) (j=0, . . . , d1) of the present and pastvalues of the differential output kact of the LAF sensor 5, the datakcmd(k−j) (=Usl(k−j), j=1, . . . , d2−1) of the past values of thetarget differential air-fuel ratio kcmd (=the SLD manipulating inputUsl) given in each control cycle from the sliding mode controller 27,and the coefficients α1, α2, βj (j=1, 2, . . . , d) calculated asdescribed above.

[0263] Then, if the set dead time d2 of the air-fuel ratio manipulatingsystem is d2=1, then the estimator 26 calculates the estimateddifferential output VO2(k+d) bar according to the equation (15), usingthe time-series data VO2(k), VO2(k−1) of the present and past values ofthe differential output VO2 of the O₂ sensor, time-series data kact(k−j)(j 32 0, . . . , d−1) of the present and past values of the differentialoutput kact of the LAF sensor 5, and the coefficients α1, α2, βj (j=1,2, . . . , d).

[0264] Then, the exhaust-side control unit 7 a calculates the SLDmanipulating input Usl (=the target differential air-fuel ratio kcmd)with the sliding mode controller 27 in STEP10.

[0265] Specifically, the sliding mode controller 27 calculates a presentvalue σ(k+d) bar (corresponding to an estimated value, after the totalset dead time d, of the linear function σ defined according to theequation (16)) of the switching function σ bar defined according to theequation (25), using the time-series data VO2(k+d) bar, VO2(k+d−1) bar(the present and preceding values of the estimated differential outputVO2 bar) of the estimated differential output VO2 bar determined by theestimator 26 in STEP9.

[0266] At this time, the sliding mode controller 27 keeps the value ofthe switching function σ bar within a predetermined allowable range. Ifthe value σ(k+d) bar determined as described above exceeds the upper orlower limit of the allowable range, then the sliding mode controller 27forcibly limits the value σ(k+d) bar to the upper or lower limit of theallowable range.

[0267] Then, the sliding mode controller 27 accumulatively adds valuesσ(k+d) bar·ΔT, produced by multiplying the present value σ(k+d) bar ofthe switching function σ bar by the period ΔT of the control cycles ofthe exhaust-side control unit 7 a. That is, the sliding mode controller27 adds the product σ(k+d) bar·ΔT of the value σ(k+d) bar and the periodΔT calculated in the present control cycle to the sum determined in thepreceding control cycle, thus calculating an integrated value σ bar(hereinafter represented by “Σσ bar”) which is the calculated result ofthe term Σ(σ bar·T) of the equation (27).

[0268] In the present embodiment, the sliding mode controller 27 keepsthe integrated value Σσ bar in a predetermined allowable range. If theintegrated value Σσ bar exceeds the upper or lower limit of theallowable range, then the sliding mode controller 27 forcibly limits theintegrated value Σσ bar to the upper or lower limit of the allowablerange.

[0269] Then, the sliding mode controller 27 calculates the equivalentcontrol input Ueq, the reaching law input Urch, and the adaptive lawinput Uadp according to the respective equations (24), (26), (27), usingthe time-series data VO2(k+d)bar, VO2(k+d−1) bar of the present and pastvalues of the estimated differential output VO2 bar determined by theestimator 26 in STEP9, the value Σ (k+d) bar of the switching function σand its integrated value Σσ bar which are determined as described above,and the gain coefficients a1, a2, b1 determined in STEP 8 (which arebasically the latest identified gain coefficients a1(k) hat, a2(k) hat,b1(k) hat).

[0270] The sliding mode controller 27 then adds the equivalent controlinput Ueq, the reaching law input Urch, and the adaptive law input Uadpto calculate the SLD manipulating input Usl, i.e., the input quantity(=the target differential air-fuel ratio kcmd) to be applied to theexhaust system E for converging the output signal VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET.

[0271] After having calculated the SLD manipulating input Usl, theexhaust-side control unit 7 a determines the stability of the adaptivesliding mode control process (or more specifically, the stability of thecontrolled state (hereinafter referred to as “SLD controlled state”) ofthe output VO2/OUT of the O₂ sensor 6 based on the adaptive sliding modecontrol process), and sets a value of a flag f/sld/stb indicative ofwhether the SLD controlled state is stable or not in STEP11. The flagf/sld/stb is “1” when the SLD controlled state is stable, and “0” whenthe SLD controlled state is not stable.

[0272] The stability determining process is carried out according to aflowchart shown in FIG. 17.

[0273] As shown in FIG. 17, the sliding mode controller 27 calculates adifference Δσ bar (corresponding to a rate of change of the switchingfunction σ bar) between the present value σ(k+d) bar of the switchingfunction σ bar calculated in STEP10 and a preceding value σ(k+d−1) barthereof in STEP11-1.

[0274] Then, the sliding mode controller 27 decides whether or not aproduct Δσ bar·σ(k+d) bar (corresponding to the time-differentiatedfunction of a Lyapunov function σ bar²/2 relative to the σ bar) of thedifference Δσ bar and the present value σ(k+d) bar of the switchingfunction σ bar is equal to or smaller than a predetermined value ε (≧0)in STEP11-2.

[0275] The product Δσ bar·σ(k+d) bar (hereinafter referred to as“stability determining parameter Pstb”) will be described below. If thestability determining parameter Pstb is greater than 0 (Pstb>0), thenthe value of the switching function σ bar is basically shifting awayfrom “0”. If the stability determining parameter Pstb is equal to orsmaller than 0 (Pstb≦0), then the value of the switching function σ baris basically converged or converging to “0”. Generally, in order toconverge a controlled variable to its target value according to thesliding mode control process, it is necessary that the value of theswitching function be stably converged to “0”. Basically, therefore, itis possible to determine whether the SLD controlled state is stable orunstable depending on whether or not the value of the stabilitydetermining parameter Pstb is equal to or smaller than 0.

[0276] If, however, the stability of the SLD controlled state isdetermined by comparing the value of the stability determining parameterPstb with “0”, then the determined result of the stability is affectedeven by slight noise contained in the value of the switching function σbar. According to the present embodiment, therefore, the predeterminedvalue ε with which the stability determining parameter Pstb is to becompared in STEP11-2 is of a positive value slightly greater than “0”.

[0277] If Pstb>ε in STEP11-2, then the SLD controlled state is judged asbeing unstable, and the value of a timer counter tm (count-down timer)is set to a predetermined initial value T_(M) (the timer counter tm isstarted) in order to inhibit the determination of the target air-fuelratio KCMD using the SLD manipulating input Usl calculated in STEP10 fora predetermined time in STEP11-4. Thereafter, the value of the flagf/sld/stb is set to “0” in STEP11-5, after which control returns to themain routine shown in FIG. 10.

[0278] If Pstb≦ε in STEP11-2, then the sliding mode controller 27decides whether the present value σ(k+d) bar of the switching function σbar falls within a predetermined range or not in STEP11-3.

[0279] If the present value σ(k+d) bar of the switching function σ bardoes not fall within the predetermined range, then since the presentvalue σ(k+d) bar is spaced widely apart from “0”, the SLD controlledstate is considered to be unstable. Therefore, if the present valueσ(k+d) bar of the switching function a bar does not fall within thepredetermined range in STEP11-3, then the SLD controlled state is judgedas being unstable, and the processing of STEP11-4 and STEP11-5 isexecuted to start the timer counter tm and set the value of the flagf/sld/stb to “0”.

[0280] In the present embodiment, since the value of the switchingfunction σ bar is limited within the allowable range in STEP10, thedecision processing in STEP11-3 may be dispensed with.

[0281] If the present value σ(k+d) bar of the switching function σ barfalls within the predetermined range in STEP11-3, then the sliding modecontroller 27 counts down the timer counter tm for a predetermined timeΔtm in STEP11-6. The sliding mode controller 27 then decides whether ornot the value of the timer counter tm is equal to or smaller than “0”,i.e., whether a time corresponding to the initial value T_(M) haselapsed from the start of the timer counter tm or not, in STEP11-7.

[0282] If tm>0, i.e., if the timer counter tm is still measuring timeand its set time has not yet elapsed, then since no substantial time haselapsed after the SLD controlled state is judged as unstable in STEP11-2or STEP11-3, the SLD controlled state tends to become unstable.Therefore, if tm>0 in STEP11-7, then the value of the flag f/sld/stb isset to “0” in STEP11-5.

[0283] If tm≦0 in STEP11-7, i.e., if the set time of the timer countertm has elapsed, then the SLD controlled stage is judged as being stable,and the value of the flag f/sld/stb is set to “1” in STEP9-8.

[0284] According to the above processing, the stability of the SLDcontrolled state is determined. If the SLD controlled state is judged asbeing unstable, then the value of the flag f/sld/stb is set to “0”, andif the SLD controlled state is judged as being stable, then the value ofthe flag f/sld/stb is set to “1”.

[0285] The above process of determining the stability of the SLDcontrolled state is by way of illustrative example only. The stabilityof the SLD controlled state may be determined by any of various otherprocesses. For example, in each given period longer than the controlcycle, the frequency with which the value of the stability determiningparameter Pstb in the period is greater than the predetermined value εis counted. If the frequency is in excess of a predetermined value, thenthe SLD controlled state is judged as unstable. Otherwise, the SLDcontrolled state is judged as stable.

[0286] Referring back to FIG. 10, after a value of the flag f/sld/stbindicative of the stability of the SLD controlled state has been set,the sliding mode controller 27 determines the value of the flagf/sld/stb in STEP12. If the value of the flag f/sld/stb is “1”, i.e., ifthe SLD controlled state is judged as being stable, then the slidingmode controller 27 limits the SLD manipulating input Usl calculated inSTEP10 in STEP13. Specifically, the sliding mode controller 27determines whether the present value Usl(k) of the SLD manipulatinginput Usl calculated in STEP10 falls in a predetermined allowable rangeor not. If the present value Usl exceeds the upper or lower limit of theallowable range, then the sliding mode controller 27 forcibly limits thepresent value Usl(k) of the SLD manipulating input Usl to the upper orlower limit of the allowable range.

[0287] The SLD manipulating input Usl (=the target differential air-fuelratio kcmd) limited in STEP13 is stored in a memory (not shown) in atime-series fashion, and will be used in the processing operation of theestimator 26.

[0288] Then, the sliding mode controller 27 adds the air-fuel ratioreference value FLAF/BASE to the SLD manipulating input Usl limited inSTEP13, thus calculating the target air-fuel ratio KCMD in STEP15. Theprocessing in the present control cycle of the exhaust-side control unit7 a is now put to an end.

[0289] If f/sld/stb=0 in STEP12, i.e., if the SLD controlled state isjudged as unstable, then the sliding mode controller 27 forcibly setsthe value of the SLD manipulating input Usl in the present control cycleto a predetermined value (the fixed value or the preceding value of theSLD manipulating input Usl) in STEP14. The sliding mode controller 27calculates the target air-fuel ratio KCMD by adding the air-fuel ratioreference value FLAF/BASE to the SLD manipulating input Usl in STEP15.The processing in the present control cycle of the exhaust-side controlunit 7 a is now put to an end.

[0290] The target air-fuel ratio KCMD finally determined in STEP15 isstored in a memory (not shown) in a time-series fashion in each controlcycle. When the general feedback controller 15 is to use the targetair-fuel ratio KCMD determined by the exhaust-side control unit 7 a (seeSTEPf in FIG. 8), the latest one of the time-series data of the targetair-fuel ratio KCMD thus stored is selected.

[0291] Details of the operation of the apparatus according to thepresent embodiment have been described above.

[0292] The operation of the apparatus will be summarized as follows: Theexhaust-side control unit 7 a sequentially determines the targetair-fuel ratio KCMD which is a target value for the upstream-of-catalystair-fuel ratio so as to converge (adjust) the output signal VO2/OUT ofthe O₂ sensor 6 downstream of the catalytic converter 3 to the targetvalue VO2/TARGET therefor. The amount of fuel injected into the internalcombustion engine 1 is adjusted to converge the output of the LAF sensor5 to the target air-fuel ratio KCMD, thereby feedback-controlling theupstream-of-catalyst air-fuel ratio at the target air-fuel ratio KCMD,and hence converging the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET. The catalytic converter 3 can thus maintain itsoptimum exhaust gas purifying performance.

[0293] In this case, in order to calculate the target air-fuel ratioKCMD according to the adaptive sliding mode control process of thesliding mode controller 27, the exhaust-side control unit 7 a uses theestimated differential output VO2 bar determined by the estimator 27,i.e., the estimated differential output VO2 bar which is an estimatedvalue of the differential output VO2 of the O₂ sensor 6 after the totalset dead time d which is the sum of the set dead time d1 of the exhaustsystem E and the set dead time d2 of the air-fuel ratio manipulatingsystem (the system comprising the internal combustion engine 1 and theengine-side control unit 7 b). The exhaust-side control unit 7 adetermines the target air-fuel ratio KCMD so as to converge theestimated value of the output VO2/OUT of the O₂ sensor 6 after the totalset dead time d which is represented by the estimated differentialoutput VO2 bar.

[0294] The estimated differential output VO2 bar determined by theestimator 26 is the estimated value of the differential output VO2 ofthe O₂ sensor 6 after the set dead times d1, d2 set by the dead timesetting means 29 depending on the estimated exhaust gas volume ABSVdetermined by the flow rate data generating means 28, i.e., the totalset dead time d determined by the set dead times d1, d2 that aresubstantially equal to the actual dead times of the exhaust system E andthe air-fuel ratio manipulating system. The algorithm for calculatingthe estimated differential output VO2 bar with the estimator 26 isconstructed on the basis of the exhaust system model and the air-fuelratio manipulating system model which have the respective dead timeelements of the set dead times d1, d2. The values of the gaincoefficients a1, a2, b1 which are parameters of the exhaust system modelare sequentially identified to minimize an error between the identifieddifferential output VO2 hat indicative of the differential output VO2 ofthe O₂ sensor 6 on the exhaust system model and the actual differentialoutput VO2, and the identified values a1 hat, a2 hat, b1 hat thereof areused in the process of calculating the estimated differential output VO2bar with the estimator 26. Since the set dead time d1 that issubstantially equal to the actual dead time of the exhaust system E isused as the dead time of the exhaust system model, the matching betweenthe exhaust system model and the behavioral characteristics of theactual exhaust system E is increased, allowing the identifier 25 todetermine the identified gain coefficients a1 hat, a2 hat, b1 hat whichaccurately reflect the actual behavior of the exhaust system E.

[0295] The estimated differential output VO2 bar determined by theestimator 26 is thus highly accurate, not depending on changes in theactual dead times of the exhaust system E and the air-fuel ratiomanipulating system, but representing the output of the O₂ sensor 6after the total dead time which is the sum of those dead times. Usingthe estimated differential output VO2 bar, the sliding mode controller27 can determine the target air-fuel ratio KCMD which is capable ofoptimally compensating for the effect of the dead times of the exhaustsystem E and the air-fuel ratio manipulating system, and hence canperform the control process of converging the output VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET accurately with a highly quickresponse. As a result, the purifying capability of the catalyticconverter 3 can be increased.

[0296] The algorithm of the adaptive sliding mode control process of thesliding mode controller 27 for determining the target air-fuel ratioKCMD is constructed on the basis of the exhaust system model having theset dead time d1 which is substantially equal to the actual dead time ofthe exhaust system E, as with the estimator 26, and uses the identifiedgain coefficients a1 hat, a2 hat, b1h hat that are sequentiallydetermined by the identifier 25 in order to determine the targetair-fuel ratio KCMD. Therefore, the target air-fuel ratio KCMD can bedetermined to as to accurately reflect the actual behavior of theexhaust system E, and the quick response of the control process ofconverging the output VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET can be increased to increase the purifying capability of thecatalytic converter 3.

[0297] The identifier 25 limits combinations of the identified gaincoefficients a1 hat, a2 hat to be determined to values within theidentifying coefficient limiting range that is variably establisheddepending on the estimated exhaust gas volume ABSV which determines theset dead times d1, d2, and also sets the value of the identified gaincoefficient b1 to a value within the range that is also variablyestablished depending on the estimated exhaust gas volume ABSV. Theidentifier 25 variably adjusts the value of the weighted parameter λ₁ inthe algorithm of the method of weighted least squares for determiningthe identified gain coefficients a1 hat, a2 hat, b1 hat, depending onthe estimated exhaust gas volume ABSV. Therefore, errors and variationsof these identified gain coefficients a1 hat, a2 hat, b1 hat can besuppressed and their reliability is increased, without depending onchanges in the actual dead times and the response delay characteristicsof the exhaust system E and the air-fuel ratio manipulating system. As aresult, the accuracy of the estimated differential output VO2 that isdetermined by the estimator 26 using the identified gain coefficients a1hat, a2 hat, b1 hat can stably be maintained, and the target air-fuelratio KCMD that is capable of converging the output VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET smoothly with a highly quickresponse can stably be determined. Thus, the high purifying capabilityof the catalytic converter 3 can stably be maintained.

[0298] A second embodiment of the present invention will be describedbelow. The present embodiment is an embodiment relating to the first andsecond aspects of the present invention, as with the above firstembodiment. The present embodiment basically differs from the previousembodiment as to only the processing operation of the estimator 26, andemploys the same reference characters as those of the previousembodiment for its description.

[0299] In the previous embodiment, the estimated value of thedifferential output VO2 of the O₂ sensor 6 after the total set dead timed (=d1+d2) is determined in order to compensate for the effect of boththe dead time d1 of the exhaust system E and the dead time d2 of theair-fuel ratio manipulating system (the system comprising the internalcombustion engine 1 and the engine-side control unit 7 b). However, ifthe dead time d2 of the air-fuel ratio manipulating system issufficiently small (it can be regarded as d2≈0) compared with the deadtime d1 of the exhaust system E, then an estimated value VO2(k+d1) bar(hereinafter referred to as “second estimated differential outputVO2(k+d1) bar”) of the differential output VO2 of the O₂ sensor 6 afterthe dead time d1 of the exhaust system E may be determined, and thetarget air-fuel ratio KCMD may be determined using the second estimateddifferential output VO2(k+d1) bar. According to the present embodiment,the second estimated differential output VO2(k+d1) bar is determined,and the output VO2/OUT of the O₂ sensor 6 is converged to the targetvalue VO2/TARGET.

[0300] The estimator 26 determines the second estimated differentialoutput VO2 bar as follows: Using the equation (1) expressing the exhaustsystem model of the exhaust system E, the second estimated differentialoutput VO2(k+d1) bar which is an estimated value VO2(k+d1) bar of thedifferential output VO2 of the O₂ sensor 6 after the dead time d1 of theexhaust system E in each control cycle is expressed by the followingequation (42), using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 and the time-series datakact(k−j) (j=1, 2, . . . , d1) of the past values of the differentialoutput kact (=KACT−FLAF/BASE) of the LAF sensor 5: $\begin{matrix}{{{{VO2}\left( {k + {d1}} \right)} = {{{\alpha 3} \cdot {{VO2}(k)}} + {{\alpha 4} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{d1}{\gamma \quad {j \cdot {{kact}\left( {k - j} \right)}}}}}}\text{where}\begin{matrix}{{{\alpha 3} = {\text{the~~first-row,~~first-column~~element~~of}\quad A^{d1}}},} \\{{{\alpha 4} = {\text{the~~first-row,~~second-column~~element~~of}\quad A^{d1}}},} \\{{\gamma \quad j} = {\text{the~~first-row~~elements~~of}\quad {A^{j - 1} \cdot B}}} \\{A = \begin{bmatrix}{a1} & {a2} \\1 & 0\end{bmatrix}} \\{B = \begin{bmatrix}{b1} \\0\end{bmatrix}}\end{matrix}} & (42)\end{matrix}$

[0301] In the equation (42), “α3”, “α4” represent the first-row,first-column element and the first-row, second-column element,respectively, of the power A^(d1) (d1: dead time of the exhaust systemE) of the matrix A defined as described above with respect to theequation (13), and “γj” (j=1, 2, . . . , d1) represents the first-rowelements of the product A^(j−1)·B of the power A^(j−1) (j=1, 2, . . . ,d1) of the matrix A and the vector B defined as described above withrespect to the equation (13).

[0302] The equation (42) is an equation for the estimator 26 tocalculate the second estimated differential output VO2(k+d1) bar. Theequation (42) is obtained from the equation (13) by settingkcmd(k)=kact(k), d=d1 (the dead time d2 of the air-fuel ratiomanipulating system is regarded as “0”) in the equation (18) describedin the first embodiment. In the present embodiment, therefore, theestimator 26 determines, in each control cycle, calculates the equation(42) to determine the second estimated differential output VO2(k+d1) barof the O₂ sensor 6, using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 and the time-series datakact(k−j) (j=1, 2, . . . , d1) of the past values of the differentialoutput kact of the LAF sensor 5.

[0303] The values of the coefficients α3, α4, γj (j=1, 2, . . . , d1)required to calculate the second estimated differential output VO2(k+d1)bar according to the equation (42) are calculated using the identifiedgain coefficients a1 hat, a2 hat, b1 hat which represent the identifiedvalues of the gain coefficients a1, a2, b1. The value of the dead timed1 required in the calculation of the equation (42) employs the set deadtime d1 that is sequentially determined in each control cycle by thedead time setting means 29, as with the first embodiment. In this case,the dead time setting means 29 is not required to determine the set deadtime d2 of the air-fuel ratio manipulating system.

[0304] Other processing details than described above are basically thesame as those of the first embodiment. However, the sliding modecontroller 27 determines the equivalent control input Ueq, the reachinglaw input Urch, and the adaptive law input Uadp, which are components ofthe SLD manipulating input Usl, according to the equations (24), (26),(27) where “d” is replaced with “d1”.

[0305] With the apparatus for controlling the air-fuel ratio of theinternal combustion engine according to the present embodiment, the setdead time d1 of the exhaust system E to be taken into account inconverging the output VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET is variably set depending on the estimated exhaust gas volumeso as to be substantially equal to the actual dead time. Using the valueof the set dead time d1, the processing sequences of the identifier 25,the estimator 26, and the sliding mode controller 27 are carried out inthe same manner as with the first embodiment. Therefore, the presentembodiment offers the same advantages as those of the first embodiment.

[0306] The apparatus for controlling the air-fuel ratio according to thepresent invention is not limited to the above embodiments, but may bemodified as follows:

[0307] In the first and second embodiments, the O₂ sensor 6 is used asthe exhaust gas sensor downstream of the catalytic converter 3. However,any of various other sensors may be employed insofar as they can detectthe concentration of a certain component of the exhaust gas down-streamof the catalytic converter to be controlled. For example, a CO sensor isemployed if the carbon monoxide (CO) in the exhaust gas downstream ofthe catalytic converter is controlled, an NOx sensor is employed if thenitrogen oxide (NOx) in the exhaust gas downstream of the catalyticconverter is controlled, and an HC sensor is employed if the hydrocarbon(HC) in the exhaust gas down-stream of the catalytic converter iscontrolled.

[0308] In the above embodiments, the differential output kact of the LAFsensor 5, the differential output VO2 of the O₂ sensor 6, and the targetdifferential air-fuel ratio kcmd are employed in the processingsequences of the identifier 25, the estimator 26, and the sliding modecontroller 27. However, the processing sequences of the identifier 25,the estimator 26, and the sliding mode controller 27 may be performeddirectly using the output KACT of the LAF sensor 5, the output VO2/OUTof the O₂ sensor 6, and the target air-fuel ratio KCMD.

[0309] In the above embodiments, the manipulated variable generated bythe exhaust-side control unit 7 a is the target air-fuel ratio KCMD (thetarget input for the exhaust system E), and the air-fuel ratio of theair-fuel mixture to be combusted by the internal combustion engine 1 ismanipulated according to the target air-fuel ratio KCMD. However, acorrected amount of the amount of fuel supplied to the internalcombustion engine 1 may be determined by the exhaust-side control unit 7a, and the amount of fuel supplied to the internal combustion engine 1may be adjusted in a feed-forward fashion from the target air-fuel ratioKCMD to manipulate the air-fuel ratio.

[0310] In the above embodiments, the sliding mode controller 27 employsan adaptive sliding mode control process which incorporates an adaptivelaw (adaptive algorithm) taking into account the effect of disturbances.However, the sliding mode controller 27 may employ a normal sliding modecontrol process which is free from such an adaptive law. Furthermore,the sliding mode controller 27 may be replaced with another type ofadaptive controller, e.g., a back-stepping controller or the like.

[0311] In the second embodiment, the control system for the air-fuelratio is constructed using the exhaust system model taking into accountthe dead time d1 of the exhaust system E. If the dead time d1 of theexhaust system E is relatively short compared with the control cycle ofthe exhaust-side control unit 7 a, for example, then a control systemmay be constructed using an exhaust system where d1=0 in the equation(1) (such a control system is concerned with the second aspect of thepresent invention). In this case, the estimator 26 is not required. Theidentifier 25 may determine the identified gain coefficients a1 hat, a2hat, b1 hat which are the identified values of the parameters of theexhaust system model according to an algorithm (the algorithm of amethod of weighted least squares) which is constructed in the samemanner as with the first embodiment where d1=0 in the equation (4). Atthis time, the weighted parameter λ₁ is sequentially variably setdepending on the estimated exhaust gas volume ABSV in the same manner aswith the first embodiment. The sliding mode controller 27 may determinethe SLD manipulating input Usl in the same manner as with the firstembodiment according to equations produced by putting d=0 in theequations (19), (20), (22), and also according to the equation (18).

Industrial Applicability

[0312] As described above, the present invention is useful forcontrolling the air-fuel ratio of an internal combustion engine mountedon an automobile or the like to increase the exhaust gas purifyingcapability of a catalytic converter.

1. An apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, identifying means for sequentially identifying the value of apredetermined parameter of a predetermined model of an exhaust system,which ranges from a position upstream of said catalytic converter tosaid exhaust gas sensor and including said catalytic converter, forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of said exhaust gas sensor via at leasta dead time element from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, manipulated variable generating meansfor sequentially generating a manipulated variable to determine anair-fuel ratio of the exhaust gas which enters said catalytic converterusing the identified value of the parameter of said model to convergethe output of said exhaust gas sensor to a predetermined target value,and air-fuel ratio manipulating means for manipulating the air-fuelratio of an air-fuel mixture to be combusted by the internal combustionengine depending on the manipulated variable, said apparatus comprising:flow rate data generating means for sequentially generating datarepresentative of a flow rate of the exhaust gas flowing through thecatalytic converter; and dead time setting means for variably setting aset dead time as the dead time of a dead time element of the model ofsaid exhaust system depending on the value of the data generated by saidflow rate data generating means, wherein said identifying meansidentifies the value of said parameter using the value of the set deadtime set by said dead time setting means.
 2. An apparatus forcontrolling the air-fuel ratio of an internal combustion engineaccording to claim 1, wherein said identifying means comprises means foridentifying the value of said parameter according to an algorithm forminimizing an error between the output of said exhaust gas sensor in themodel of said exhaust system and an actual output of said exhaust gassensor, said apparatus further comprising: means for variably settingthe value of a weighted parameter of said algorithm depending on thevalue of the data generated by said flow rate data generating means. 3.An apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, identifying means for sequentially identifying the value of apredetermined parameter of a predetermined model of an exhaust system,which ranges from a position upstream of said catalytic converter tosaid exhaust gas sensor and including said catalytic converter, forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of said exhaust gas sensor from theair-fuel ratio of the exhaust gas which enters said catalytic converter,manipulated variable generating means for sequentially generating amanipulated variable to determine an air-fuel ratio of the exhaust gaswhich enters said catalytic converter using the identified value of theparameter of said model to converge the output of said exhaust gassensor to a predetermined target value, and air-fuel ratio manipulatingmeans for manipulating the air-fuel ratio of an air-fuel mixture to becombusted by the internal combustion engine depending on the manipulatedvariable, wherein: said identifying means comprises means foridentifying the value of said parameter according to an algorithm forminimizing an error between the output of said exhaust gas sensor in themodel of said exhaust system and an actual output of said exhaust gassensor; wherein said apparatus further comprising: flow rate datagenerating means for sequentially generating data representative of aflow rate of the exhaust gas flowing through the catalytic converter;and means for variably setting the value of a weighted parameter of thealgorithm of said identifying means depending on the value of the datagenerated by said flow rate data generating means.
 4. An apparatus forcontrolling the air-fuel ratio of an internal combustion engineaccording to any one of claim 1 through 3, wherein identifying meansdetermines the identified value of the parameter of the model of saidexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data generated by saidflow rate data generating means.
 5. A method of controlling the air-fuelratio of an internal combustion engine, comprising the steps ofsequentially identifying the value of a predetermined parameter of apredetermined model of an exhaust system, which ranges from a positionupstream of a catalytic converter disposed in an exhaust passage of theinternal combustion engine to an exhaust gas sensor disposed downstreamof the catalytic converter for detecting the concentration of aparticular component in an exhaust gas, and includes said catalyticconverter, for expressing a behavior of the exhaust system which isregarded as a system for generating the output of said exhaust gassensor via at least a dead time element from the air-fuel ratio of theexhaust gas which enters said catalytic converter, and sequentiallygenerating a manipulated variable to determine an air-fuel ratio of theexhaust gas which enters said catalytic converter using the identifiedvalue of the parameter of said model in order to converge the output ofsaid exhaust gas sensor to a predetermined target value, said internalcombustion engine having air-fuel ratio manipulating means formanipulating the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine depending on the manipulated variable,said method comprising the steps of: sequentially generating datarepresentative of a flow rate of the exhaust gas flowing through thecatalytic converter, and sequentially setting a set dead time as thedead time of a dead time element of the model of said exhaust systemdepending on the value of the data representative of the flow rate ofthe exhaust gas, wherein said step of identifying the parameter of themodel of said exhaust system identifies the value of said parameterusing the value of said set dead time.
 6. A method of controlling theair-fuel ratio of an internal combustion engine according to claim 5,wherein said step of identifying the parameter of the model of saidexhaust system identifies the value of said parameter according to analgorithm for minimizing an error between the output of said exhaust gassensor in the model of said exhaust system and an actual output of saidexhaust gas sensor, and variably sets the value of a weighted parameterof said algorithm depending on the value of the data generated by saidflow rate data generating means.
 7. A method of controlling the air-fuelratio of an internal combustion engine, comprising the steps ofsequentially identifying the value of a predetermined parameter of apredetermined model of an exhaust system, which ranges from a positionupstream of a catalytic converter disposed in an exhaust passage of theinternal combustion engine to an exhaust gas sensor disposed downstreamof the catalytic converter for detecting the concentration of aparticular component in an exhaust gas, and includes said catalyticconverter, for expressing a behavior of the exhaust system which isregarded as a system for generating the output of said exhaust gassensor from the air-fuel ratio of the exhaust gas which enters saidcatalytic converter, and sequentially generating a manipulated variableto determine an air-fuel ratio of the exhaust gas which enters saidcatalytic converter using the identified value of the parameter of saidmodel in order to converge the output of said exhaust gas sensor to apredetermined target value, wherein the air-fuel ratio of an air-fuelmixture to be combusted by the internal combustion engine is manipulateddepending on the manipulated variable, wherein: said step of identifyingthe parameter of the model of said exhaust system comprises the step ofidentifying the value of said parameter according to an algorithm forminimizing an error between the output of said exhaust gas sensor in themodel of said exhaust system and an actual output of said exhaust gassensor; said method further comprising the steps of: sequentiallygenerating data representative of a flow rate of the exhaust gas flowingthrough the catalytic converter, and variably setting the value of aweighted parameter of said algorithm for identifying the parameter ofsaid model depending on the value of the data representative of the flowrate of the exhaust gas.
 8. A method of controlling the air-fuel ratioof an internal combustion engine according to any one of claims 5through 7, wherein said step of identifying the parameter of the modelof said exhaust system determines the identified value of the parameterof the model of said exhaust system by limiting the identified value toa value within a predetermined range depending on the value of the datarepresentative of the flow rate of the exhaust gas.
 9. A recordingmedium readable by a computer and storing an air-fuel ratio controlprogram for enabling said computer to perform a process of sequentiallyidentifying the value of a predetermined parameter of a predeterminedmodel of an exhaust system, which ranges from a position upstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine to an exhaust gas sensor disposed downstream of thecatalytic converter for detecting the concentration of a particularcomponent in an exhaust gas, and includes said catalytic converter, forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of said exhaust gas sensor via at leasta dead time element from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, a process of sequentially generating amanipulated variable to determine an air-fuel ratio of the exhaust gaswhich enters said catalytic converter using the identified value of theparameter of said model in order to converge the output of said exhaustgas sensor to a predetermined target value, and a process ofmanipulating the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine depending on the manipulated variable,wherein said program comprising: said air-fuel ratio control programincludes a program for enabling the computer to perform a process ofsequentially generating data representative of a flow rate of theexhaust gas flowing through the catalytic converter, and sequentiallysetting a value of said set dead time as the dead time of a dead timeelement of the model of said exhaust system depending on the value ofthe data representative of the flow rate of the exhaust gas, and saidprogram for enabling the computer to identify the parameter of the modelof said exhaust system identifies the parameter using the value of saidset dead time.
 10. A recording medium storing an air-fuel ratio controlprogram for an internal combustion engine according to claim 9, whereinthe program of said air-fuel ratio control program for enabling thecomputer to perform the process of identifying the value of theparameter of the model of said exhaust system identifies the value ofsaid parameter according to an algorithm for minimizing an error betweenthe output of said exhaust gas sensor in the model of said exhaustsystem and an actual output of said exhaust gas sensor, and variablysets the value of a weighted parameter of said algorithm depending onthe value of the data representative of the flow rate of said exhaustgas.
 11. A recording medium readable by a computer and storing anair-fuel ratio control program for enabling said computer to perform aprocess of sequentially identifying the value of a predeterminedparameter of a predetermined model of an exhaust system, which rangesfrom a position upstream of a catalytic converter disposed in an exhaustpassage of the internal combustion engine to an exhaust gas sensordisposed downstream of the catalytic converter for detecting theconcentration of a particular component in an exhaust gas, and includessaid catalytic converter, for expressing a behavior of the exhaustsystem which is regarded as a system for generating the output of saidexhaust gas sensor from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, a process of sequentially generating amanipulated variable to determine an air-fuel ratio of the exhaust gaswhich enters said catalytic converter using the identified value of theparameter of said model in order to converge the output of said exhaustgas sensor to a predetermined target value, and a process ofmanipulating the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine depending on the manipulated variable,wherein: the program of said air-fuel ratio control program for enablingthe computer to perform the process of identifying the value of theparameter of the model of said exhaust system identifies the value ofsaid parameter according to an algorithm for minimizing an error betweenthe output of said exhaust gas sensor in the model of said exhaustsystem and an actual output of said exhaust gas sensor; and saidair-fuel ratio control program includes a program for enabling thecomputer to perform a process of sequentially generating-datarepresentative of a flow rate of the exhaust gas flowing through thecatalytic converter, and a process of variably setting the value of aweighted parameter of said algorithm for identifying the parameter ofsaid model depending on the value of the data representative of the flowrate of the exhaust gas.
 12. A recording medium storing an air-fuelratio control program for an internal combustion engine according to anyone of claims 9 through 11, wherein the program of said air-fuel ratiocontrol program for enabling the computer to perform the process ofidentifying the value of the parameter of the model of said exhaustsystem determines the identified value of the parameter of the model ofsaid exhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data representative ofthe flow rate of the exhaust gas.